Calculus II - 24
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The Comparison Test allows determining if a series converges or diverges by comparing it to standard p-series or geometric series. If a series is less than or equal to a convergent p-series, or greater than or equal to a divergent p-series, then the series has the same behavior. Examples demonstrate applying the test to various series.
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Calculus II - 25
This document discusses tests for determining if infinite series converge or diverge, including:
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A power series is a series of the form Σan(x-c)n where an are the coefficients of the series. The sum of the series is a function of x. A power series centered at c or about c is of the form Σan(x-c)n. For a given power series, there are three possibilities for convergence: it converges only at c, converges for all x, or converges when |x-c|<R and diverges when |x-c|>R, where R is the radius of convergence.
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This document discusses tests for determining if infinite series converge or diverge, including:
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2) Absolute and conditional convergence, where a series is absolutely convergent if the sum of the absolute values converges, and conditionally convergent if the series converges but the sum of absolute values does not.
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The document introduces the integral test, which can be used to determine if an infinite series is convergent or divergent. The integral test states that a series is convergent if the corresponding improper integral converges, and divergent if the improper integral diverges. Several examples demonstrate applying the integral test to various series by setting up and evaluating the corresponding improper integrals. The document also discusses using integrals to estimate the sum of a convergent series.
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There are three types of conic sections: parabolas, ellipses, and hyperbolas. A parabola is the set of points equidistant from a fixed point (focus) and line (directrix). An ellipse is the set of points where the sum of distances from two fixed points (foci) is a constant. A hyperbola is the set of points where the difference of distances from two fixed points (foci) is a constant. Equations of parabolas, ellipses, and hyperbolas can be written and their properties like foci, vertices and asymptotes can be determined from the equations.
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The cross product and dot product of vectors are introduced. Properties of the cross product include that it is only defined for 3D vectors and that the magnitude of the cross product equals the area of the parallelogram determined by the two vectors. Vector, parametric, and various forms of plane equations are presented along with examples of finding equations of lines and planes given conditions.
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1) A sequence is an ordered list of numbers written as {an}. A sequence converges to a limit L if the terms an get arbitrarily close to L as n increases.
2) Common properties of convergent sequences include that the sum, difference, and scalar multiple of two convergent sequences is also convergent.
3) Useful theorems include if {an} converges to L and {bn} converges to M, then {an + bn} converges to L + M. If |an| ≤ M for all n and M is a constant, then the sequence converges.
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The document discusses arc length, curvature, and formulas for calculating them for curves in 2D and 3D. It provides the definitions of arc length, unit tangent vector, and curvature. Examples are given of finding the curvature of various curves, such as circles, helices, and parabolas, at given points using parametrizations of the curves. In general, the curvature of a plane curve f(x) = y is k(x) = f''(x) / [1+(f'(x))^2]^(3/2).
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The document discusses Taylor series and Maclaurin series. It provides the formulas for determining the coefficients of a Taylor series centered at a point a. It also discusses Taylor's inequality which relates the remainder of a Taylor series to the derivative of the function. Several examples are worked out of finding the Maclaurin series for common functions and proving they represent the functions for all values. Additional power series are derived from fundamental series like e^x, sin(x), and cos(x).
Review of series
This document summarizes various tests that can be used to determine if an infinite series converges or diverges, including:
1) The divergence test, integral test, p-series test, comparison test, limit comparison test, alternating series test, and ratio test.
2) It also discusses power series, including determining the radius of convergence and using Taylor series approximations with Taylor's inequality to estimate the remainder term.
3) Key concepts are that convergence tests check if partial sums approach a limit, while divergence tests examine the behavior of individual terms, and that power series have a radius of convergence determining the interval on which they converge.
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1. The document discusses Chi-square tests for categorical variables, including tests for goodness of fit, comparing proportions between independent samples, and contingency tables.
2. Examples are provided to illustrate Chi-square tests for comparing the results of two medical testing methods and evaluating whether patient outcomes are independent of a treatment.
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Calculus II - 28
The document discusses Taylor series and how they can be used to represent general functions as power series. It provides the formulas for determining the coefficients in the Taylor series expansion of a function f(x) about a point a. Examples are given of finding the Maclaurin series, which is the Taylor series expansion about x=0, for common functions and determining their radii of convergence. The key steps in proving that the Taylor series equals the original function within the radius of convergence are outlined using Taylor's inequality.
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The dot product of two vectors a and b is defined as a·b and is calculated by multiplying their corresponding components and adding them. It can be used to calculate the projection of one vector onto another and the cosine of the angle between the vectors. The cross product of two vectors a and b is defined as axb and produces a third vector that is perpendicular to both a and b. It has various properties including being anticommutative and having a magnitude equal to the area of the parallelogram determined by the two vectors.
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Cochran's Q test is a nonparametric test used to compare dependent categorical variables across three or more conditions. It is an extension of the McNemar test for related samples. The test compares the probability of a target response across different conditions to see if treatments have identical effects. It assumes a large sample size and that blocks were randomly selected. An example uses data from pet stores to test if they are equally likely to display both reptiles (snakes and lizards) during different times of the year. The Cochran's Q test rejects the null hypothesis, finding stores were more likely to display both during Christmas than other times.
Power series & Radius of convergence
The document uses the ratio test to find the radius of convergence for two power series. For the first series, the ratio test gives an interval of convergence from -2 to 4, so the radius of convergence is 3. For the second series, the ratio test gives the interval of convergence as from -infinity to infinity, so the radius of convergence is infinite.
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The document defines power series as sums of terms involving constants multiplied by powers of x, where x can be centered at 0 or another constant a. It states the convergence theorem for power series: if a power series converges at a point c ≠ 0, then it converges absolutely for all x within |c|, and if it diverges at a point d, then it diverges for all x outside |d|. It introduces the radius of convergence R, where a power series converges absolutely for |x-a|<R and diverges for |x-a|>R, regardless of what happens exactly at R.
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The document discusses calculating the area of a surface obtained by rotating a curve around an axis. It provides the general formulas for finding the area which involves integrating the function squared plus the integral of its derivative. Several examples are worked out for rotating curves and arcs around the x-axis and y-axis. The document also gives an example problem of finding the force on a dam due to hydrostatic water pressure.
Calculus II - 9
The document discusses the arc length formula and how to calculate arc length. The arc length formula is the integral of the square root of (dx/dt)^2 + (dy/dt)^2 from t1 to t2, where t1 and t2 are the starting and ending points. Examples are provided on calculating the perimeter of a circle using this formula and approximating the arc length of a hyperbola between two points.
Calculus II - 8
This document discusses different types of improper integrals. It defines Type I improper integrals as integrals where the integrand is defined at every point but the limit at one or both endpoints is infinite. Type II integrals have an integrand that is continuous on some interval excluding one or both endpoints. The document provides examples of evaluating various improper integrals and checking whether they converge or diverge. It also introduces a comparison theorem for improper integrals.
Calculus II - 7
This document discusses numerical integration techniques for approximating integrals of functions that have no closed-form antiderivative. It introduces the midpoint rule, trapezoidal rule, and Simpson's rule for approximating integrals using Riemann sums over subintervals. An example applies Simpson's rule to approximate a definite integral and estimates the error bound.
More from David Mao (12)
Calculus II - 24
- 1. 11.4 The Comparison Tests The convergence of some series can be determined easily by the integral test, for example: = Some series look similar, but the integral test cannot be easily applied, for example: = +
- 2. The Comparison Test: ∞ ∞ Suppose = and = are series with positive terms, ∞ If = is convergent and ≤ ∞ then = is also convergent. ∞ If = is divergent and ≥ ∞ then = is also divergent. If = > is finite then they either both converge or both diverge.
- 3. Common series used for comparison: p-series: converges for > = diverges for = geometric series: converges for | | < = diverges for | | =
- 4. Ex: Test the series for convergence. = +
- 5. Ex: Test the series for convergence. = + < + is convergent. = So is convergent. = +
- 6. Ex: Test the series for convergence. =
- 7. Ex: Test the series for convergence. = > is divergent. = So is divergent. =
- 8. Ex: Test the series for convergence. =
- 9. Ex: Test the series for convergence. = = is convergent. = So is convergent. =
- 10. + Ex: Test the series for convergence. = +
- 11. + Ex: Test the series for convergence. = + + = + is divergent (p-test). = + So is divergent. = +
- 12. + Ex: Test the series for convergence. = +
- 13. + Ex: Test the series for convergence. = + + = + is convergent (geometric). = + So is convergent. = +
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