This document analyzes a quasi-Fibonacci-like polynomial sequence defined by the recursion Fn = Fn-1 + x^2Fn-2, with initial conditions F0 = -1 and F1 = x - 1. It proves several properties of this sequence, including: (1) the maximum root of Fn, denoted gn, is monotonically increasing for odd n; (2) all roots except possibly one are irrational; (3) F2n has no real roots for all n. It also analyzes the existence and behavior of the roots as n increases.
A Proof of Hypothesis Riemann and it is proven that apply only for equation ζ(z)=0.Also here it turns out that it does not apply in General Case.(DOI:10.13140/RG.2.2.10888.34563/5)
This document provides an overview of factoring polynomials. It begins by explaining that factoring polynomials is an important skill, as it is often the first step in solving problems in later chapters. It then defines factoring as determining what was multiplied to obtain a given quantity. Several methods of factoring polynomials are described, including: factoring out the greatest common factor; factoring by grouping; and factoring quadratic polynomials into two linear factors. Step-by-step examples are provided to illustrate each method.
The 1741 Goldbach [1] made his most famous contribution to mathematics with the conjecture that all even numbers can be expressed as the sum of the two primes (currently Conjecture) referred to as “all even numbers greater than 2 can be expressed as the sum-two primes” (DOI:10.13140/RG.2.2.32893.69600/1)
It turns out that an equation x^p+y^q=z^w has a solution in Z+ if and only if at least one exponent p or q or w equals 2.DOI:10.13140/RG.2.2.15459.22567/7
1. To solve equations with radicals, isolate the radical term by adding or subtracting like terms, then square both sides of the equation.
2. The "u" substitution method can be used to solve quadratic equations in radical form by letting the variable under the radical equal "u" and squaring it.
3. After substitution and factoring, the solutions for "u" are then substituted back in terms of the original variable under the radical.
The document discusses several theorems related to twin prime conjectures:
- Theorem 1 states that a prime p can be written in the form 3k+1 or 3k-1, with k being even.
- Theorem 4 characterizes twin primes as pairs where n(n+2) satisfies a modular condition.
- Theorems aim to prove there are infinitely many twin primes, relying on the ratio of primes increasing without bound as n increases without bound.
1) Primes are positive integers greater than 1 that are only divisible by 1 and themselves. The fundamental theorem of arithmetic states that every positive integer can be uniquely expressed as the product of primes.
2) Euclid's proof shows there are infinitely many primes. Euclid numbers form a sequence where each term is the sum of the previous terms plus 1, and the early terms are prime. However, not all Euclid numbers are prime.
3) The largest power of a prime p that divides n! is given by the sum of the number of times p divides the numbers from 1 to n in their prime factorizations. This can be determined from the number of 1s in the binary representation
The document discusses applications of factoring polynomials. It provides examples of how factoring can be used to evaluate polynomials by substituting values into the factored form. Factoring is also useful for determining the sign of outputs and for solving polynomial equations, which is described as the most important application of factoring. Examples are given to demonstrate evaluating polynomials both with and without factoring, and checking the answers obtained from factoring using the expanded form.
A Proof of Hypothesis Riemann and it is proven that apply only for equation ζ(z)=0.Also here it turns out that it does not apply in General Case.(DOI:10.13140/RG.2.2.10888.34563/5)
This document provides an overview of factoring polynomials. It begins by explaining that factoring polynomials is an important skill, as it is often the first step in solving problems in later chapters. It then defines factoring as determining what was multiplied to obtain a given quantity. Several methods of factoring polynomials are described, including: factoring out the greatest common factor; factoring by grouping; and factoring quadratic polynomials into two linear factors. Step-by-step examples are provided to illustrate each method.
The 1741 Goldbach [1] made his most famous contribution to mathematics with the conjecture that all even numbers can be expressed as the sum of the two primes (currently Conjecture) referred to as “all even numbers greater than 2 can be expressed as the sum-two primes” (DOI:10.13140/RG.2.2.32893.69600/1)
It turns out that an equation x^p+y^q=z^w has a solution in Z+ if and only if at least one exponent p or q or w equals 2.DOI:10.13140/RG.2.2.15459.22567/7
1. To solve equations with radicals, isolate the radical term by adding or subtracting like terms, then square both sides of the equation.
2. The "u" substitution method can be used to solve quadratic equations in radical form by letting the variable under the radical equal "u" and squaring it.
3. After substitution and factoring, the solutions for "u" are then substituted back in terms of the original variable under the radical.
The document discusses several theorems related to twin prime conjectures:
- Theorem 1 states that a prime p can be written in the form 3k+1 or 3k-1, with k being even.
- Theorem 4 characterizes twin primes as pairs where n(n+2) satisfies a modular condition.
- Theorems aim to prove there are infinitely many twin primes, relying on the ratio of primes increasing without bound as n increases without bound.
1) Primes are positive integers greater than 1 that are only divisible by 1 and themselves. The fundamental theorem of arithmetic states that every positive integer can be uniquely expressed as the product of primes.
2) Euclid's proof shows there are infinitely many primes. Euclid numbers form a sequence where each term is the sum of the previous terms plus 1, and the early terms are prime. However, not all Euclid numbers are prime.
3) The largest power of a prime p that divides n! is given by the sum of the number of times p divides the numbers from 1 to n in their prime factorizations. This can be determined from the number of 1s in the binary representation
The document discusses applications of factoring polynomials. It provides examples of how factoring can be used to evaluate polynomials by substituting values into the factored form. Factoring is also useful for determining the sign of outputs and for solving polynomial equations, which is described as the most important application of factoring. Examples are given to demonstrate evaluating polynomials both with and without factoring, and checking the answers obtained from factoring using the expanded form.
The document is an advertisement for Vedantu, an online education platform, promoting their free online admission test to win scholarships for classes 6-12, JEE, and NEET. It highlights success stories of students who scored well in board exams and engineering/medical entrance exams after taking online classes on Vedantu. It encourages students to register now for the admission test to secure limited seats and chance at 100% scholarship.
1) The document discusses synthetic division, which is a method for dividing polynomials without using variables.
2) It provides an example of using synthetic division to determine if 1 is a root of the polynomial 4x - 3x + x + 5.
3) Another example uses synthetic division to find the quotient and remainder of (4x - 7x - 11x + 5) divided by (4x - 5).
This module introduces polynomial functions of degree greater than 2. It covers identifying polynomial functions from relations, determining the degree of a polynomial, finding quotients of polynomials using division algorithm and synthetic division, and applying the remainder and factor theorems. The document provides examples and practice problems for each topic. It aims to teach students how to work with higher degree polynomial functions.
The security of the RSA algorithm depends on the difficulty of factoring large numbers. The best known factoring algorithms are trial division, Dixon's algorithm, the quadratic sieve, and the number field sieve. The quadratic sieve and number field sieve are parallelizable algorithms that improve on Dixon's algorithm by using a "sieving" technique to more efficiently find relations between factors. While factoring performance improves incrementally over time, a large key size (over 300 bits) is still considered secure against the best known factoring methods.
Division of polynomials follows the same rules as division of real numbers. To divide a polynomial by a monomial, each term of the dividend is divided by the monomial divisor. To divide a polynomial by a polynomial, long division is used by repeatedly determining the quotient and remainder until the division is complete. The steps of long division are shown through an example of dividing a third degree polynomial by a linear polynomial divisor. The quotient and remainder are checked by multiplying the divisor and quotient together.
GR 8 Math Powerpoint about Polynomial Techniquesreginaatin
-This is a powerpoint inspired by one of Canva displayed presentation.
- This is about Math Polynomials and good for highschoolers presentation for school.
- It consists of 39 pages explaining each of the Polynomial Techniques.
- Good for review or inspired powerpoint.
Rational expressions are expressions of the form P/Q, where P and Q are polynomials. Polynomials are expressions of the form anxn + an-1xn-1 + ... + a1x1 + a0. Rational expressions can be written in either expanded or factored form. The factored form is useful for determining the domain of a rational expression, solving equations involving rational expressions, evaluating inputs, and determining the sign of outputs. The domain of a rational expression excludes values of x that make the denominator equal to 0.
The document provides examples of finding the vertex of parabolic functions by completing the square. It shows working through examples where the coefficient of x^2 is 1 and where it is not 1. In both cases, the process involves separating terms containing x, squaring half the coefficient of x, and factorizing the resulting expression to put it in the vertex form of f(x)=(x-h)^2+k, where (h,k) gives the vertex.
The document provides a step-by-step guide for finding the vertex of parabolic functions by completing the square. It gives two examples, finding that the vertex of f(x)=x^2 -4x+3 is (2,-1) and the vertex of f(x)=-2x^2 -2x+1 is (-1/2,-1). Completing the square involves factoring the quadratic term and rearranging constants to put the function in vertex form f(x)=a(x-h)^2 + k, where (h,k) gives the vertex coordinates.
1. Graph E represents the graph where b > 0.5.
2. Graph D represents the graph where x < -7.7.
3. Solving the equation 6(x + 3) + 1 = -11 yields x = -7.
Notes solving polynomials using synthetic divisionLori Rapp
This document discusses using synthetic division to evaluate polynomials at specific values and factor polynomials. It provides examples of using synthetic division to:
1) Evaluate polynomials like f(x) = x^2 - x + 5 at specific values such as f(-2)
2) Factor polynomials when one factor is known, such as factoring x^3 - 3x^2 - 13x + 15 after determining (x + 3) is a factor
3) Find all zeros of a polynomial by setting each factor equal to 0 after factoring
Step-by-step instructions and additional examples are provided to illustrate the process.
The document discusses using synthetic division to evaluate polynomials at specific values and factor polynomials. It provides examples of using synthetic division to:
1) Evaluate polynomials like f(x) = x^2 - x + 5 at specific values such as f(-2).
2) Factor polynomials when one factor is known, such as factoring x^3 - 3x^2 - 13x + 15 after determining (x + 3) is a factor using synthetic division.
3) Determine if a binomial is a factor of a polynomial, such as showing (x - 3) is a factor of x^3 + 4x^2 - 15x - 18.
The document discusses techniques for combining fractions with opposite denominators. It explains that we can multiply the numerator and denominator by -1 to change the denominator to its opposite. It provides examples of switching fractions to their opposite denominators and combining fractions with opposite denominators by first switching one denominator so they are the same. It also discusses an alternative approach of pulling out a "-" from the denominator and passing it to the numerator when switching denominators, ensuring the leading term is positive for polynomial denominators.
I have added to the original presentation in response to one of the comments.... the result of 'x' is correct on slide 7, take a look at the new version of this ppt to clear up any confusion about why...
The document is a lesson on implicit differentiation and related concepts:
1) Implicit differentiation allows one to take the derivative of an implicitly defined relation between x and y, even if y is not explicitly defined as a function of x.
2) Examples are provided to demonstrate implicit differentiation, such as finding the slope of a tangent line to a curve.
3) The van der Waals equation is introduced to describe non-ideal gas properties, and implicit differentiation is used to find the isothermal compressibility of a van der Waals gas.
The document discusses methods for finding the general solution to linear differential equations of second order with constant coefficients. It presents four types of complementary functions depending on whether the roots of the auxiliary equation are real and distinct, real and equal, complex, or surd roots. It also describes four types of particular integrals depending on whether the given function is an exponential, sine, cosine, or contains an exponential term. The document provides examples of solving differential equations of each type and includes multiple choice questions to test understanding of the concepts and methods presented.
The document discusses synthetic division and factoring polynomials. It provides an example of using synthetic division to divide one polynomial by another. Key points covered include: the remainder theorem, which states that the remainder of dividing a polynomial P(x) by (x-c) is equal to the value of P(c); and the factor theorem, which states that c is a zero of P(x) if and only if (x-c) is a factor of P(x). The document then provides an example of using the fact that P(1)=0 to factor a polynomial completely.
The document discusses terms, factors, and cancellation in mathematics expressions. It defines a term as one or more quantities that are added or subtracted, and a factor as a quantity that is multiplied to other quantities. Cancellation can be used to simplify fractions by canceling common factors in the numerator and denominator. However, terms cannot be canceled as they represent distinct quantities being added or subtracted. Several examples demonstrate identifying terms and factors and applying cancellation when possible.
Mat221 5.6 definite integral substitutions and the area between two curvesGlenSchlee
This document contains examples of evaluating definite integrals using substitution and finding the area between curves. It includes 8 examples of evaluating definite integrals using techniques like u-substitution, integration by parts, and trigonometric substitutions. It also contains 3 examples of finding the area of regions bounded by graphs by setting up and evaluating definite integrals with respect to x or y.
This document provides examples of solving linear equations by manipulating the equations to isolate the variable. It shows adding or subtracting the same quantity from both sides, distributing terms, combining like terms, and dividing both sides by a coefficient to isolate the variable. The examples are worked through step-by-step and checked by substitution back into the original equation.
This document analyzes the diophantine equation x^2 + y^2 = n^2 in connection with a famous problem proposed at the 1988 International Mathematical Olympiad. It defines a function F(x,y) and uses properties of F to find integer solutions to the equation. It shows that for prime numbers p, the only integer solutions are the trivial ones (0,p^2) and (p^3,p^2). More generally, it finds that for any number n, the only integer solutions are n^2 or possibly another non-trivial integer if n belongs to a specific sequence generated by another number. It concludes by conjecturing the full set of integer solutions.
This document provides an introduction to large deviations theory through examples and theorems. It begins with coin tossing experiments and computes the decay of probabilities that the empirical mean differs from the expected value. Specifically, it shows the limit of the log probability approaches a rate function I(x) defined piecewise. The document then considers sums of independent normal and Bernoulli random variables. Finally, it introduces Cramer's theorem, which generalizes the results to any independent identically distributed random variables using the cumulant generating function.
The document is an advertisement for Vedantu, an online education platform, promoting their free online admission test to win scholarships for classes 6-12, JEE, and NEET. It highlights success stories of students who scored well in board exams and engineering/medical entrance exams after taking online classes on Vedantu. It encourages students to register now for the admission test to secure limited seats and chance at 100% scholarship.
1) The document discusses synthetic division, which is a method for dividing polynomials without using variables.
2) It provides an example of using synthetic division to determine if 1 is a root of the polynomial 4x - 3x + x + 5.
3) Another example uses synthetic division to find the quotient and remainder of (4x - 7x - 11x + 5) divided by (4x - 5).
This module introduces polynomial functions of degree greater than 2. It covers identifying polynomial functions from relations, determining the degree of a polynomial, finding quotients of polynomials using division algorithm and synthetic division, and applying the remainder and factor theorems. The document provides examples and practice problems for each topic. It aims to teach students how to work with higher degree polynomial functions.
The security of the RSA algorithm depends on the difficulty of factoring large numbers. The best known factoring algorithms are trial division, Dixon's algorithm, the quadratic sieve, and the number field sieve. The quadratic sieve and number field sieve are parallelizable algorithms that improve on Dixon's algorithm by using a "sieving" technique to more efficiently find relations between factors. While factoring performance improves incrementally over time, a large key size (over 300 bits) is still considered secure against the best known factoring methods.
Division of polynomials follows the same rules as division of real numbers. To divide a polynomial by a monomial, each term of the dividend is divided by the monomial divisor. To divide a polynomial by a polynomial, long division is used by repeatedly determining the quotient and remainder until the division is complete. The steps of long division are shown through an example of dividing a third degree polynomial by a linear polynomial divisor. The quotient and remainder are checked by multiplying the divisor and quotient together.
GR 8 Math Powerpoint about Polynomial Techniquesreginaatin
-This is a powerpoint inspired by one of Canva displayed presentation.
- This is about Math Polynomials and good for highschoolers presentation for school.
- It consists of 39 pages explaining each of the Polynomial Techniques.
- Good for review or inspired powerpoint.
Rational expressions are expressions of the form P/Q, where P and Q are polynomials. Polynomials are expressions of the form anxn + an-1xn-1 + ... + a1x1 + a0. Rational expressions can be written in either expanded or factored form. The factored form is useful for determining the domain of a rational expression, solving equations involving rational expressions, evaluating inputs, and determining the sign of outputs. The domain of a rational expression excludes values of x that make the denominator equal to 0.
The document provides examples of finding the vertex of parabolic functions by completing the square. It shows working through examples where the coefficient of x^2 is 1 and where it is not 1. In both cases, the process involves separating terms containing x, squaring half the coefficient of x, and factorizing the resulting expression to put it in the vertex form of f(x)=(x-h)^2+k, where (h,k) gives the vertex.
The document provides a step-by-step guide for finding the vertex of parabolic functions by completing the square. It gives two examples, finding that the vertex of f(x)=x^2 -4x+3 is (2,-1) and the vertex of f(x)=-2x^2 -2x+1 is (-1/2,-1). Completing the square involves factoring the quadratic term and rearranging constants to put the function in vertex form f(x)=a(x-h)^2 + k, where (h,k) gives the vertex coordinates.
1. Graph E represents the graph where b > 0.5.
2. Graph D represents the graph where x < -7.7.
3. Solving the equation 6(x + 3) + 1 = -11 yields x = -7.
Notes solving polynomials using synthetic divisionLori Rapp
This document discusses using synthetic division to evaluate polynomials at specific values and factor polynomials. It provides examples of using synthetic division to:
1) Evaluate polynomials like f(x) = x^2 - x + 5 at specific values such as f(-2)
2) Factor polynomials when one factor is known, such as factoring x^3 - 3x^2 - 13x + 15 after determining (x + 3) is a factor
3) Find all zeros of a polynomial by setting each factor equal to 0 after factoring
Step-by-step instructions and additional examples are provided to illustrate the process.
The document discusses using synthetic division to evaluate polynomials at specific values and factor polynomials. It provides examples of using synthetic division to:
1) Evaluate polynomials like f(x) = x^2 - x + 5 at specific values such as f(-2).
2) Factor polynomials when one factor is known, such as factoring x^3 - 3x^2 - 13x + 15 after determining (x + 3) is a factor using synthetic division.
3) Determine if a binomial is a factor of a polynomial, such as showing (x - 3) is a factor of x^3 + 4x^2 - 15x - 18.
The document discusses techniques for combining fractions with opposite denominators. It explains that we can multiply the numerator and denominator by -1 to change the denominator to its opposite. It provides examples of switching fractions to their opposite denominators and combining fractions with opposite denominators by first switching one denominator so they are the same. It also discusses an alternative approach of pulling out a "-" from the denominator and passing it to the numerator when switching denominators, ensuring the leading term is positive for polynomial denominators.
I have added to the original presentation in response to one of the comments.... the result of 'x' is correct on slide 7, take a look at the new version of this ppt to clear up any confusion about why...
The document is a lesson on implicit differentiation and related concepts:
1) Implicit differentiation allows one to take the derivative of an implicitly defined relation between x and y, even if y is not explicitly defined as a function of x.
2) Examples are provided to demonstrate implicit differentiation, such as finding the slope of a tangent line to a curve.
3) The van der Waals equation is introduced to describe non-ideal gas properties, and implicit differentiation is used to find the isothermal compressibility of a van der Waals gas.
The document discusses methods for finding the general solution to linear differential equations of second order with constant coefficients. It presents four types of complementary functions depending on whether the roots of the auxiliary equation are real and distinct, real and equal, complex, or surd roots. It also describes four types of particular integrals depending on whether the given function is an exponential, sine, cosine, or contains an exponential term. The document provides examples of solving differential equations of each type and includes multiple choice questions to test understanding of the concepts and methods presented.
The document discusses synthetic division and factoring polynomials. It provides an example of using synthetic division to divide one polynomial by another. Key points covered include: the remainder theorem, which states that the remainder of dividing a polynomial P(x) by (x-c) is equal to the value of P(c); and the factor theorem, which states that c is a zero of P(x) if and only if (x-c) is a factor of P(x). The document then provides an example of using the fact that P(1)=0 to factor a polynomial completely.
The document discusses terms, factors, and cancellation in mathematics expressions. It defines a term as one or more quantities that are added or subtracted, and a factor as a quantity that is multiplied to other quantities. Cancellation can be used to simplify fractions by canceling common factors in the numerator and denominator. However, terms cannot be canceled as they represent distinct quantities being added or subtracted. Several examples demonstrate identifying terms and factors and applying cancellation when possible.
Mat221 5.6 definite integral substitutions and the area between two curvesGlenSchlee
This document contains examples of evaluating definite integrals using substitution and finding the area between curves. It includes 8 examples of evaluating definite integrals using techniques like u-substitution, integration by parts, and trigonometric substitutions. It also contains 3 examples of finding the area of regions bounded by graphs by setting up and evaluating definite integrals with respect to x or y.
This document provides examples of solving linear equations by manipulating the equations to isolate the variable. It shows adding or subtracting the same quantity from both sides, distributing terms, combining like terms, and dividing both sides by a coefficient to isolate the variable. The examples are worked through step-by-step and checked by substitution back into the original equation.
This document analyzes the diophantine equation x^2 + y^2 = n^2 in connection with a famous problem proposed at the 1988 International Mathematical Olympiad. It defines a function F(x,y) and uses properties of F to find integer solutions to the equation. It shows that for prime numbers p, the only integer solutions are the trivial ones (0,p^2) and (p^3,p^2). More generally, it finds that for any number n, the only integer solutions are n^2 or possibly another non-trivial integer if n belongs to a specific sequence generated by another number. It concludes by conjecturing the full set of integer solutions.
This document provides an introduction to large deviations theory through examples and theorems. It begins with coin tossing experiments and computes the decay of probabilities that the empirical mean differs from the expected value. Specifically, it shows the limit of the log probability approaches a rate function I(x) defined piecewise. The document then considers sums of independent normal and Bernoulli random variables. Finally, it introduces Cramer's theorem, which generalizes the results to any independent identically distributed random variables using the cumulant generating function.
The document contains solutions to several exercises involving proofs of properties related to Fibonacci numbers, Lucas numbers, and Zeckendorf representations. It proves that f2n = fn2 + 2fn-1fn for positive integers n using mathematical induction. It finds the first 12 Lucas numbers and proves that Lm+n = fm+1Ln + fmLn-1 for positive integers m and n>1. It also finds the Zeckendorf representations of several integers and defines a recursive formula for negative Fibonacci numbers.
This document contains the solution to a problem involving a sequence of continuously differentiable functions defined by a recurrence relation. The solution shows that:
1) The sequence is monotonically increasing and bounded, so it converges pointwise to a limit function g(x).
2) The limit function g(x) is the unique fixed point of the operator defining the recurrence, and is equal to 1/(1-x).
3) Uniform convergence on compact subsets is proved using Dini's theorem and properties of the operator.
The document contains solutions to 4 problems posed at the IMC 2016 conference in Bulgaria.
The first problem proves that the sum of a sequence of positive numbers divided by increasing powers of 2 is less than or equal to 2. The second problem finds the minimum value of a function over continuous functions satisfying a given inequality.
The third problem proves that if a function satisfies three properties related to permutations, then the size of the ring it is defined over must be congruent to 2 modulo 4.
The fourth problem proves an inequality relating the number of integer solutions to an inequality when the upper bound is increased or decreased by 1.
ECE 302 Spring 2012 covers practice problems involving various continuous and discrete random variables, including uniform, exponential, normal, lognormal, Rayleigh, Cauchy, Pareto, Gaussian mixture, Erlang, and Laplace distributions. The document provides an example problem solving a uniform random variable. It also lists some suggested reading materials and references textbooks for further information.
This document presents concepts related to the convergence of the Bernoulli distribution to other probability distributions. It begins with definitions of the Bernoulli distribution and introduces binomial, Poisson, and Gaussian distributions. It then presents three lemmas to be used in proofs in the next section. The lemmas cover the limiting behavior of exponential terms, factorials, and an approximation for large factorials. The document aims to directly prove convergence results, unlike most previous works that used an indirect approach via moment generating functions.
This document presents concepts related to the convergence of the Bernoulli distribution to other probability distributions. It begins with definitions of the Bernoulli distribution and introduces binomial, Poisson, and Gaussian distributions. It then presents three lemmas to be used in proofs in the next section. The lemmas cover the limiting behavior of exponential terms, factorials, and an approximation for large factorials. The document aims to directly prove convergence results, unlike most previous works that used an indirect approach via moment generating functions.
This paper presents concepts of Bernoulli distribution, and how it can be used as an approximation of Binomial, Poisson, and Gaussian distributions with a different approach from earlier existing literature. Due to discrete nature of the random variable X, a more appropriate method of the principle of mathematical induction (PMI) is used as an alternative approach to limiting behavior of the binomial random variable. The study proved de Moivre–Laplace theorem (convergence of binomial distribution to Gaussian distribution) to all values of p such that p ≠ 0 and p ≠ 1 using a direct approach which opposes the popular and most widely used indirect method of moment generating function.
The document discusses Fourier series and their applications. It begins by introducing how Fourier originally developed the technique to study heat transfer and how it can represent periodic functions as an infinite series of sine and cosine terms. It then provides the definition and examples of Fourier series representations. The key points are that Fourier series decompose a function into sinusoidal basis functions with coefficients determined by integrating the function against each basis function. The series may converge to the original function under certain conditions.
The document provides an overview of the integral test for determining convergence or divergence of infinite series.
1) It introduces the integral test theorem, which states that if f is continuous, positive, and decreasing, the series of terms f(n) converges if and only if the integral of f from 1 to infinity converges.
2) Examples are provided to illustrate applying the integral test to test convergence of specific series, such as 1/n^2 and ln(n)/n.
3) It defines p-series as having terms of the form 1/n^p and uses the integral test to prove such series converge if p>1 and diverge if p≤1.
This document discusses three methods for finding the roots of nonlinear equations:
1) Bisection method, which converges linearly but is guaranteed to find a root.
2) Newton's method, which converges quadratically (much faster) but may diverge if the starting point is too far from the root.
3) Secant method, which is faster than bisection but slower than Newton's, and also requires starting points close to the root. Newton's and secant methods can be extended to systems of nonlinear equations.
This document contains the answers to exercises for the third edition of the textbook "Microeconomic Analysis" by Hal R. Varian. The answers are organized by chapter and include solutions to mathematical problems as well as explanations and justifications. Key information provided in the answers includes derivations of production functions, profit functions, cost functions, and factor demand functions for various technologies. Convexity and monotonicity properties of technologies are also analyzed.
The document discusses Fourier series and periodic functions. It provides:
- Definitions of Fourier series and periodic functions.
- Examples of periodic functions including trigonometric and other functions.
- Integration properties used to solve Fourier series problems.
- Two examples showing the steps to obtain the Fourier series of given periodic functions in a specified interval.
The document discusses Fourier series and periodic functions. It provides:
- Definitions of Fourier series and periodic functions.
- Examples of periodic functions including trigonometric and other functions.
- Euler's formulae for calculating the coefficients of a Fourier series.
- Integration properties used to solve Fourier series problems.
- Two examples of determining the Fourier series for given periodic functions and using it to deduce mathematical results.
This document provides information about derivatives and their applications:
1. It defines the derivative as the limit of the difference quotient, and explains how to calculate derivatives using first principles. It also covers rules for finding derivatives of sums, products, quotients, exponentials, and logarithmic functions.
2. Higher order derivatives are introduced, with examples of how to take second and third derivatives.
3. Applications of derivatives like finding velocity and acceleration from a position-time function are demonstrated. Maximum/minimum values and how to find local and absolute extrema are also discussed with an example.
This document contains lecture notes on evaluating definite integrals. It introduces the definition of the definite integral as a limit of Riemann sums, and properties of integrals such as additivity and comparison properties. It also states the Second Fundamental Theorem of Calculus, which relates definite integrals to indefinite integrals via the derivative of the integrand function. Examples are provided to illustrate how to use these properties and theorems to evaluate definite integrals.
The document introduces numerical methods for finding the roots or zeros of equations of the form f(x) = 0, where f(x) is an algebraic or transcendental function. It focuses on the bisection method, also called the Bolzano method, which uses interval bisection to bracket the root between two values where f(x) has opposite signs. The method iteratively narrows down the interval to find the root to within a specified tolerance. Several examples demonstrate applying the bisection method to find roots of polynomial, logarithmic, and trigonometric equations.
The document discusses mathematical induction and recursive definitions. It provides examples of using induction to prove statements for all natural numbers, like n < 2n. It also gives examples of recursively defined sequences, functions, and sets, such as the Fibonacci numbers defined by f(n) = f(n-1) + f(n-2). Recursive definitions define an object in terms of itself, similar to induction which proves statements by showing that if true for n, then true for n+1.
1. Remarks on the Properties of a Quasi-Fibonacci-like
Polynomial Sequence
Brice Merwine
LIU Brooklyn
Ilan Weinschelbaum
Wesleyan University
Abstract
Consider the Quasi-Fibonacci-like Polynomial Sequence given by F0 = −1, F1 =
x − 1 and for n ≥ 2, Fn = Fn−1 + x2Fn−2. Denote the maximum root of Fn by gn. In
this article, we will analyze the existence and nonexistence of gn as well as study the
behavior of the sequence {gn}. We will prove all but one of the roots are irrational and
the sequence of the maximum odd-indexed roots are monotonically increasing.
1 Introduction
Consider the well-studied Fibonacci Polynomial Sequence,
Fn = xFn−1 + Fn−2, for n ≥ 2 with F0 = 0, F1 = 1.
Many results are known about this polynomial sequence. It is known that Fn(1) is the nth
Fibonacci number. Hogatt and Bicknell [HB] also gave an explicit form for the zeros of these
polynomials.
Further work includes Molina and Zeleke [MZ] generalizing of the initial conditions and
exploring the recursion,
Fn = xk
Fn−1 + Fn−2,
now known as the Fibonacci-like polynomials. They made a number of discoveries about the
asymptotic behavior of the roots of these polynomials.
This inspired further work by Brandon Alberts in 2011. Alberts studied the Quasi-
Fibonacci polynomials. These are polynomials defined by the following recursion:
Fq
n = Fq
n−1 + xk
Fq
n−2, for n ≥ 2 with Fq
0 = −1, Fq
1 = x − 1, where k = 1.
He found a number of interesting results including the existence of all roots and convergence
of all roots to the same value, namely 2. He found that the roots of the even-indexed
polynomials converged from below and the roots of the odd-indexed polynomials converged
from above.
In this paper, we study a modified recursion, a Quasi-Fibonacci-Like polynomial se-
quence, where k = 2. Specifically, this recursion is as follows:
Fq
n = Fq
n−1 + x2
Fq
n−2, for n ≥ 2 with Fq
0 = −1, Fq
1 = x − 1.
1
2. We will explore the existence and nonexistence of the roots, as well as their behavior as a
sequence. We will also numerically and computationally examine the asymptotic behavior of
these roots in addition to showing all but one root are irrational. We denote the maximum
root of Fq
n as gn, and for the sake of simplicity, we suppress the q superscript for the rest of
this paper.
2 Formulas and technical results
The formulas we will use throughout this paper are as follows:
Lemma 1.
Fn(x) =
∞
i=0
n−i−1
i
x2i+1
− n−i
i
x2i
(1)
F2n(x) = −
n
i=0
2n−i
i
x2i
+
n−1
i=0
2n−i−1
i
x2i+1
(2)
F2n+1(x) =
n
i=0
2n−i
i
x2i+1
− 2n+1−i
i
x2i
(3)
Fn(x) =
(x − 1 + λ−)λn
+ − (x − 1 + λ+)λn
−
λ+ − λ−
, where λ± =
1 ±
√
1 + 4x2
2
. (4)
Fn(x) = (1 + 2x2
)Fn−2(x) − x4
Fn−4(x) (5)
Remark. Formula (4) is known at the Binet Form.
Proof. Formula (1) We proceed by induction. We begin by showing the base cases.
Case 1: n = 1
∞
i=0
1 − i − 1
i − 1
(−1) +
1 − i − 1
i
(x − 1) x2i
=
0
−1
(−1) +
0
0
(x − 1) x0
+
−1
0
(−1) +
−1
1
(x − 1) x2
+ . . .
= [0 · (−1) + 1 · (x − 1)] · 1 + 0 + 0 + . . .
= x − 1 = F1(x)
Case 2: n = 2
∞
i=0
2 − i − 1
i − 1
(−1) +
2 − i − 1
i
(x − 1) x2i
=
1
−1
(−1) +
1
0
(x − 1) x0
+
0
0
(−1) +
0
1
(x − 1) x2
+ . . .
= [0 · (−1) + 1 · (x − 1)] · 1 + [1 · (−1) + 0 · (x − 1)]x2
+ 0 + . . .
= −x2
+ x − 1 = F2(x)
2
3. We now show the inductive step. Suppose this identity holds for all n < m. Then
Fm(x) = Fm−1(x) + x2
Fm−2(x)
=
∞
i=0
m − i − 2
i − 1
(−1) +
m − i − 2
i
(x − 1) x2i
+ x2
∞
i=0
m − i − 3
i − 1
(−1) +
m − i − 3
i
(x − 1) x2i
=
∞
i=0
m − i − 2
i − 1
(−1) +
m − i − 2
i
(x − 1) x2i
+
∞
j=1
m − j − 2
j − 2
(−1) +
m − j − 2
j − 1
(x − 1) x2j
.
Reindexing and combining terms we have
Fm(x) =
∞
i=0
m−i−2
i−1
+ m−i−2
i−2
(−1) + m−i−2
i
+ m−i−2
i−1
(x − 1) x2i
=
∞
i=0
m−i−1
i−1
(−1) + m−i−1
i
(x − 1) x2i
giving Formula (1).
Formulas (2) and (3) follow directly from Formula (1).
Formula (4): This is the Binet Form for this recursion. We proceed by using the stan-
dard method of obtaining a Binet Formula. We establish the following system of equations:
c1λ0
+ + c2λ0
− = F0(x)
c1λ+ + c2λ− = F1(x)
where λ+ and λ− are the solutions to the equation λ2
= λ + x2
. From this quadratic
equation, we find λ+ and λ− to be as in Formula (4). Solving the system above, we find
c1 =
x − 1 + λ−
λ+ − λ−
and c2 =
− (x − 1 + λ+)
λ+ − λ−
.
Formula (5): This follows from direct manipulation of the recursion. Indeed we have
Fn(x) = Fn−1(x) + x2
Fn−2(x)
= (1 + 2x2
)Fn−2 − x2
Fn−2 + x2
Fn−3
= (1 + 2x2
)Fn−2 − x4
Fn−4.
Lemma 2. For all x < 0, we have 0 > Fn(x) > Fn+1(x) for all n.
3
4. Proof. When x < 0, F1(x) = x − 1 < −1 = F0(x) < 0. Furthermore, supposing this
statement holds up to Fn−1, we see that since Fn−2(x) < 0, Fn(x) = Fn−1(x) + x2
Fn−2(x) <
Fn−1(x).
Lemma 3. Fn(0) = −1 for all n.
Proof. Fn(0) will be the constant term of Fn; according to our Formula (1), this will be
− n−0
0
= −1
Lemma 4. limx→∞ F2n+1(x) = ∞ for all n.
Proof. First we show this is satisfied for initial values of n. Proceeding by induction we have
the base cases
lim
x→∞
F1 = lim
x→∞
x − 1 = ∞
and
lim
x→∞
F3 = lim
x→∞
x3
− 2x2
+ x − 1 = ∞.
Suppose we’ve shown this for F1, F3, ..., F2n−1. By using Formula (3), we have
lim
x→∞
F2n+1(x) = lim
x→∞
n
i=0
− 2n−i
i−1
+ 2n−i
i
(x − 1) x2i
.
Since the end behavior of a polynomial is determined by the sign of the coefficient on the
highest degree, it is sufficient to show that this coefficient is positive.
Note that 2n−i
i
= 0 when 2n − i > i. However the leading coefficient must be nonzero,
so 2n − i ≤ i. This implies n ≤ i . Examining the term with the highest degree, we see
−
2n − (n)
(n) − 1
+
2n − (n)
n
(x − 1) x2(n)
= −
n
n − 1
+
n
n
(x − 1) x2n
= −
n
n − 1
+ x − 1 x2n
= −
n
n − 1
x2n
+ x2n+1
− x2n
.
Thus the coefficient of the term with highest degree, namely x2n+1
, is positive and there-
fore
lim
x→∞
F2n+1 = ∞.
Lemma 5. limx→∞ F2n(x) = −∞ , for all n.
Proof. Lemma 5 can be proven using similar methods to Lemma 4.
4
5. 3 Existence and nonexistence of the roots
Lemma 6. For any x and any n ≥ 1, 0 > F2n(x) ≥ F2n+2(x). In particular, F2n(x) has no
real roots.
Proof. When x ≤ 0, this statement follows directly from Lemmas 2 and 3. So let x > 0. We
now proceed by induction, starting with the base cases. Notice that
F2(x) = −x2
+ x − 1 < 0
and
F2(x) − F4(x) = x4
− 2x3
+ 2x2
= x2
((x − 1)2
+ 1) > 0,
so 0 > F2(x) > F4(x) for all real values of x.
We now show the inductive step. Suppose now that we have shown this property through
F2n−2.
Case 1: x ≤
√
2. Then
2x2
F2n−2(x) ≤ x4
F2n−2(x) ≤ x4
F2n−4(x).
From Formula 5, we have
F2n(x) = (1 + 2x2
)F2n−2(x) − x4
F2n−4(x) ≤ F2n−2(x).
Case 2: x >
√
2 > 0. Recall c1 = x − 1 + λ−
λ+−λ−
and c2 = −(x − 1 + λ+)
λ+−λ−
.
0 < x
0 > −4x
4x2
− 4x + 1 < 1 + 4x2
(2x − 1)2
< 1 + 4x2
Taking the primary root of both sides provides,
(2x − 1) −
√
1 + 4x2 < 0
c1 < 0.
A similar proof shows c2 < 0. Now we will show 1 − λ+ − x2
< 0.
√
2 < x
0 < x2
− 2
0 < 4x4
− 8x2
1 + 4x2
< 4x4
− 4x2
+ 1
Taking the primary root of both sides provides
1 − 2x2
+
√
1 + 4x2 < 0
1 − λ+ − x2
< 0.
A similar proof shows 1 − λ− − x2
< 0.
5
6. Since c1, c2, (1 − λ+ − x2
), and (1 − λ− − x2
) are all negative, from Formula 4 giving the
Binet form of Fn, we have
F2n−2 − F2n = c1λ2n−2
+ + c2λ2n−2
− − c1λ2n
+ − c2λ2n
−
= c1λ2n−2
+ (1 − λ+ − x2
) + c2λ2n−2
− (1 − λ− − x2
)
> 0
Thus F2n−2 > F2n so that for all x, F2n(x) is negative and therefore has no real roots.
Lemma 7. F2n+1 has at least one real root.
Proof. Notice that the leading coefficient of F2n+1 is positive and F2n+1(0) = −1.
By the intermediate value theorem, F2n+1 has at least one real root.
4 Main Results
For the remainder of this paper, let gn denote the maximum real root of Fn.
Theorem 1. 1 = g1 < g3 < · · · < g2n+1 for all n.
Proof. We proceed by induction. Starting with the base cases direct computation shows
1 = g1 < g3. We now show the induction step. Suppose we have shown this through
g2n−1. Then since g2n−3 is a maximum root and limx→∞ F2n−3(x) = +∞, we note that
F2n−3(g2n−1) > 0. So
F2n+1(g2n−1) = (1 + 2g2
2n−1) · F2n−1(g2n−1) − g4
2n−1 · F2n−3(g2n−1)
= −g4
2n−1 · F2n−3(g2n−1) < 0.
Thus, since lim
x→∞
F2n+1(x) = +∞, F2n+1 must have a root g2n+1 > g2n−1.
Theorem 2. The roots of F2n+1 are irrational for n > 0.
Proof. If there is some rational number r which is a root of F2n+1 then, by the Rational Root
Theorem and Formula 3,
r = ±
−
2n − n
n − 1
+
2n − n
n
−
2n
−1
+
2n
0
= ±(−n + 1).
Case 1: r = −n + 1. By Theorem 1, g2n+1 > 1. However for all n, we have r ≤ 1. So
−n + 1 cannot be a root.
6
7. Case 2: r = n−1. We have shown this cannot be a root for n < 62 by direct computation.
Through manipulation of Formula (3) we have
F2n+1(x) =
n+1
i=0
−
2n − i
i − 1
+
2n − i
i
(x − 1) x2i
.
When substituting r into the expression above for x, it is easy to show that only the coefficient
of the last term is negative. It is given by
(−2)(n − 1)2n
.
Finding the ratio between the original binomial coefficients provides,
2n − i
i − 1
=
2n − i
i
i
2n − 2i + 1
.
Since the last term is the only negative term, it is sufficient to show that the third to last
term is larger. Using i = n − 2 with n ≥ 62 gives,
n + 2
n − 1
n + 1
n − 1
n
n − 1
4n
5
−
8
5
≥ 48.
(n − 2)(n + 1)(n)(n − 1)
24
4n
5
−
8
5
≥ 2(n − 1)4
2n − (n − 2)
n − 2
−(n − 2)
2n − 2(n − 2) + 1
+ n − 2 (n − 1)2(n−2)
≥ 2(n − 1)2n
where the left hand side is the third to last term in the above summation. Therefore, g2n+1
is irrational for all n > 0.
5 Numerical evidence
Our research has also suggested certain other results. One such possibility is that the odd-
indexed polynomials, F2n+1, have exactly one real root.
As the following graph shows, the first few odd-indexed polynomials have exactly one
real root.
7
8. Thus we have constructed the following lemma and conjecture.
Lemma 8. F2n+1 has no real roots on (−∞, g2n−1] ∪ (g2n+1, ∞).
Proof. This is obvious for F1(x) = x − 1. Suppose we have shown it through F2n−1.
When x > g2n+1, this follows directly from maximality of g2n+1. Therefore, let x ≤ g2n−1.
Recall that F2n(x) < 0 and note that F2n−1(x) ≤ 0. Then
F2n+1(x) = F2n(x) + x2
F2n−1(x) < 0
The previous lemma does not exclude the possibility of a root existing on the interval
(g2n−1, g2n+1), providing the following conjecture formed through observation.
Conjecture 1. F2n+1 has no real roots on (g2n−1, g2n+1).
This has been confirmed by direct calculation through F599, meaning all of these poly-
nomials have exactly one real root. It is our hope that future work may be able to formally
prove this for all odd-indexed Fn.
Our work has also led us to believe the following conjecture:
Conjecture 2. The sequence {g2n+1} is unbounded.
The roots do not appear to asymptotically approach any number through g599, selected
roots are shown below.
8
9. n gn
1 1
3 1.755
5 2.402
13 4.616
29 8.390
101 22.544
233 44.969
419 73.762
599 100.05
Graphically, we can also see that they do not appear to asymptotically approach any
number. After multiple regression analyses, we can say that it appears to grow slightly
faster than logarithmically.
6 Acknowledgements
We would like to thank Michigan State University and the Lyman Briggs College for their
support of our REU, as well as our faculty mentor, Dr. Aklilu Zeleke. We would also like to
thank his graduate assistants, Justin Droba, Rani Satyam and Richard Shadrach. Project
sponsored by the National Security Agency under Grant Number H98230-13-1-0259. Project
sponsored by the National Science Foundation under Grant Number DMS 1062817. Special
thanks to Daniel Thompson.
9
10. 7 References
• Brandon Alberts, On the Properties of a Quasi-Fibonacci Polynomial Sequence, preprint,
SURIEM 2011
• V. E. Hoggatt, Jr. and Marjorie Bicknell, Roots of Fibonacci polynomials, Fibonacci
Quart., 11(3):271-274, 1973.
• Robert Molina and Aklilu Zeleke, Generalizing results on the convergence of the maxi-
mum roots of Fibonacci type polynomials, In Proceedings of the Fortieth Southeastern
International Conference on Combinatorics, Graph Theory and Computing, volume
195, pages 95-104, 2009.
10