Ch11review
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This document contains 24 review questions about key concepts in chapter 11 of a calculus textbook, including: 1) The definition of convergence for sequences and using it to prove limits. 2) The definitions of convergence for series, absolute convergence, and conditional convergence. 3) Tests for determining if series converge or diverge, including the nth term test. 4) Power series, including radius of convergence, intervals where series converge absolutely or conditionally. 5) Taylor series and polynomials, Taylor's formula and remainder theorem, and using series to approximate functions and evaluate limits.
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Problem 1
Recall that 1
1−x =
∑∞
n=0 x
n for |x| < 1.
Find a power series representation for the following functions and state the radius of
convergence for the power series
a) f(x) = x
2
(1+x)2
.
b) f(x) = 2
1+4x2
.
c) f(x) = x
4
2−x.
d) f(x) = x
1+x2
.
e) f(x) = 1
6+x
.
f) f(x) = x
2
27−x3 .
Problem 2
Find a Taylor series with a = 0 for the given function and state the radius of conver-
gence. You may use either the direct method (definition of a Taylor series) or known
series.
a) f(x) = ln(1 + x)
b) f(x) = sin x
x
c) f(x) = x sin(3x).
Problem 3
Find the radius of convergence and interval of convergence for the series
∑∞
n=1
(x+2)n
n4n
.
Ans. Radius r=2,
√
2 − 2 < x <
√
2 + 2. Problem 4
Find the interval of convergence of the following power series. You must justify your
answers.
∑∞
n=0
n2(x+4)n
23n
.
Ans. −12 < x < 4.
Problem 5
For the function f(x) = 1/
√
x, find the fourth order Taylor polynomial with a=1.
Problem 6
A curve has the parametric equations
x = cos t, y = 1 + sin t, 0 ≤ t ≤ 2π
a) Find dy
dx
when t = π
4
.
b) Find the equation of the line tangent to the curve at t = π/4. Write it in y = mx+b
form.
c) Eliminate the parameter t to find a cartesian (x, y) equation of the curve.
d) Using (c), or otherwise identify the curve.
Problem 7
State whether the given series converges or diverges
a)
∑∞
n=0 (−1)
n+1 n22n
n!
.
b)
∑∞
n=0
n(−3)n
4n−1
.
c)
∑∞
n=1
sin n
2n2+n
.
Problem 8
1
Approximate the value of the integral
∫ 1
0
e−x
2
dx with an error no greater than 5×10−4.
Ans.
∫ 1
0
e−x
2
dx = 1 − 1
3
+ 1
5.2!
− 1
7.3!
+ ... +
(−1)n
(2n+1)n!
+ .... n ≥ 5,
for n=5
∫ 1
0
e−x
2
dx ≈ 1 − 1
3
+ 1
5.2!
− 1
7.3!
+ 1
9.4!
− 1
11.5!
≈ 0.747.
Problem 9
Find the radius of convergence for the series
∑∞
n=1
nn(x−2)2n
n!
.
Ans. R = 1√
e
.
Problem 10
Let f(x) =
∑∞
n=0
(x−1)n
n2+1
.
a) Calculate the domain of f.
b) Calculate f ′(x).
c) Calculate the domain of f ′.
Problem 11
Let f(x) =
∑∞
n=0
cos n
n!
xn.
a) Calculate the domain of f.
b) Calculate f ′(x).
c) Calculate
∫
f(x)dx.
Problem 12
Using properties of series, known Maclaurin expansions of familiar functions and their
arithmetic, calculate Maclaurin series for the following.
a) ex
2
b) sin 2x
c)
∫
x5 sin xdx
d) cos x−1
x2
e)
d((x+1) tan−1(x))
dx
Problem 13
Calculate the Taylor polynomial T5(x), expanded at a=0, for
f(x) =
∫ x
0
ln |sect + tan t|dt.
Ans. T5(x) =
x2
2
+ x
4
4!
.
Problem 14
Suppose we only consider |x| ≤ 0.8. Find the best upper bound or maximum value
you can for∣∣∣sin x − (x − x33! + x55! )∣∣∣
Same question: If
(
x − x
3
3!
+ x
5
5!
)
is used to approximate sin x for |x| ≤ 0.8. What is
the maximum error? Explain what method you are using.
Problem 15
The Taylor polynomial T5(x) of degree 5 for (4 + x)
3/2 is
(4 + x)3/2 ≈ 8 + 3x + 3
16
x2 − 1
128
x3 + 3
4096
x4 − 3
32768
x5.
a) Use this polynomial to find Taylor polynomials for (4 + ...
Ch05 1
This document discusses power series and their properties. It defines convergent and absolutely convergent power series, and introduces the ratio test to determine the radius of convergence of a power series. Examples are provided to demonstrate how to find the radius of convergence and interval of convergence. The relationship between power series and Taylor series is explained. Analytic functions are defined as those with a Taylor series representation. Methods for shifting the index of summation in power series are demonstrated.
Section 11.8
The document discusses power series and their intervals of convergence. It provides examples of using the ratio test to determine the radius of convergence and interval of convergence for various power series. Specifically, it finds:
1) The interval of convergence for ∞∑n=1 (x - 3)n/n is [2,4);
2) The interval of convergence for ∞∑n=0 n!xn is {0}; and
3) The radius of convergence is 3 and interval of convergence is (-5,1) for the power series ∞∑n=0 n(x+2)n/3n+1.
Nbhm m. a. and m.sc. scholarship test 2006
This document provides instructions for a 150-minute mathematics scholarship test consisting of 45 multiple-choice questions across three sections: Algebra, Analysis, and Geometry. The instructions specify that candidates should answer each question in the provided answer booklet and not on the question paper. Various mathematical notations and concepts are defined for reference in answering the questions.
Notes on Intersection theory
Notes on intersection theory written for a seminar in Bonn in 2010.
Following Fulton's book the following topics are covered:
- Motivation of intersection theory
- Cones and Segre Classes
- Chern Classes
- Gauss-Bonet Formula
- Segre classes under birational morphisms
- Flat pull back
Engr 371 final exam april 1996
This document contains a final exam with 7 multiple part questions testing concepts in statistics, probability, and signal processing. The exam allows standard calculators and requires showing all work. Questions cover topics like distributions, confidence intervals, Monte Carlo integration, random processes, and sampling. Students are advised that reasonable assumptions can be made if they have difficulties with a part of a question. The exam is worth a total of 80 marks plus 4 bonus marks.
Cs229 cvxopt
This document provides an overview of convex optimization. It begins by explaining that convex optimization can efficiently find global optima for certain functions called convex functions. It then defines convex sets as sets where linear combinations of points in the set are also in the set. Common examples of convex sets include norm balls and positive semidefinite matrices. Convex functions are defined as functions where linear combinations of points on the graph lie below the line connecting those points. Convex functions have properties like their first and second derivatives satisfying certain inequalities, allowing efficient optimization.
Infinite series-Calculus and Analytical Geometry
It includes all the infinite series in your course. It includes all the formulas required to solve the problem.
MTH101 - Calculus and Analytical Geometry- Lecture 44
Virtual University
Course MTH101 - Calculus and Analytical Geometry
Lecture No 44
Instructor's Name: Dr. Faisal Shah Khan
Course Email: mth101@vu.edu.pk
GATE Mathematics Paper-2000
This document contains a mathematics exam with 30 multiple choice questions covering various topics in calculus, linear algebra, differential equations, real analysis, and probability. The exam has 75 total marks and is divided into sections on vectors and matrices, differential equations, real analysis, and probability. It provides the questions, response options, and asks test takers to select the correct option(s) and write them in the answer booklet.
1624 sequence
The document defines sequences and series, and methods for determining if series converge or diverge. It provides examples of several types of series:
- Geometric series, which converge if the common ratio r is between -1 and 1, and diverge if r is outside this range.
- P-series, which converge if the exponent p is greater than 1 and diverge if p is less than or equal to 1.
- Alternating series, which may converge if the terms approach zero and alternate in sign.
Tests covered include the integral test, direct comparison test, limit comparison test, and alternating series test, which can determine if a series converges or diverges.
Nbhm m. a. and m.sc. scholarship test 2012 with answer key
The document provides instructions for a mathematics scholarship test consisting of 30 questions across three sections: Algebra, Analysis, and Geometry. It outlines the structure and time limit of the test, how to answer questions, notations that will be used, and clarifies that calculators are not allowed. The key at the end provides the answers to sample questions asked in each section to illustrate the nature and difficulty of the test.
Set convergence
This document defines key concepts related to set convergence, including:
1) The inner and outer limits of a sequence of sets in topological and normed spaces, which describe the limit inferior and limit superior of the sets.
2) Properties of set convergence like the limit inferior and limit superior being characterized as intersections and unions of the sets over cofinal subsets of the natural numbers.
3) Characterizations of the inner and outer limits of sets in terms of open neighborhoods in topological spaces and open balls in normed spaces.
4) The inner limit of a sequence of sets in a normed space being the points for which the distance to the sets goes to zero as the index increases.
draft5
This document summarizes the BTW sandpile model on a square lattice and defines relevant concepts. It describes how sandpiles are represented as height functions on the lattice, stable configurations, toppling rules, addition of sandpiles, and the group structure of recurrent sandpiles. Algorithms to find the identity of this group are developed in later sections.
Section 11.5
This document discusses alternating series and absolute convergence. It defines alternating series as series whose terms are alternately positive and negative. The alternating series test provides criteria for determining if an alternating series converges, namely if the terms decrease to zero. Absolute convergence means the corresponding series of absolute values converges, while conditional convergence means a series converges but is not absolutely convergent. Examples are provided to demonstrate determining if series are absolutely convergent, conditionally convergent, or divergent.
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Say speak talk tell
SA Y SPEAK T ALK TELL
-Speak thường dùng khi 1 người nói với 1 tập thể
-Talk thường dùng khi 2 hay nhiều người đối thoại với nhau
-Say theo sau bởi words (cấu trúc: say something to somebody)
-Tell thường dùng để truyền tải thông tin (cấu trúc: tell somebody something)
Cụ thể:
. SAY:
Là động từ có tân ngữ, có nghĩa là”nói ra, nói rằng”, chú trọng nội dung được nói ra. Ex:
- Please say it again in English. (Làm ơn nói lại bằng tiếng Anh).
- They say that he is very ill. (Họ nói rằng cậu ấy ốm nặng).
. SPEAK:
Có nghĩa là “nói ra lời, phát biểu”, chú trọng mở miệng, nói ra lời.
Thường dùng làm động từ không có tân ngữ và cũng thường sử dụng với một giới từ ‘to‘, ‘about‘ hoặc ‘of‘ trước một tân ngữ. Khi có tân ngữ thì chỉ là một số ít từ chỉ thứ tiếng”truth” (sự thật).
Ex:
- He is going to speak at the meeting. (Anh ấy sẽ phát biểu trong cuộc mít tinh).
- I speak Chinese. I don’t speak Japanese. (Tôi nói tiếng Trung Quốc. Tôi không nói tiếng Nhật Bản).
- She spoke about her work at the university.
Bà ta nói về thành tích của mình tại trường Đại học.
- He spoke of his interest in photography.
Anh ta nói về sở thích về nhiếp ảnh.
Khi muốn “nói với ai” thì dùng speak to sb hay speak with sb.
Ex:
- She is speaking to our teacher. (Cô ấy đang nói chuyện với thày giáo của chúng ta).
. TELL:
Có nghĩa “kể, chú trọng, sự trình bày”. Thường gặp trong các kết cấu : tell sb sth (nói với ai điều gì đó), tell sb to do sth (bảo ai làm gì), tell sb about sth (cho ai biết về điều gì). Ex:
- The teacher is telling the class an interesting story. (Thầy giáo đang kể cho lớp nghe một câu chuyện thú vị).
- Please tell him to come to the blackboard. (Làm ơn bảo cậu ấy lên bảng đen).
- We tell him about the bad news. (Chúng tôi nói cho anh ta nghe về tin xấu đó).
. TALK:
Có nghĩa là”trao đổi, chuyện trò”, có nghĩa gần như speak, chú trọng động tác “nói’. Thường gặp trong các kết cấu: talk to sb (nói chuyện với ai), talk about sth (nói về điều gì), talk with sb (chuyện trò với ai).
Ex:
- What are they talking about? (Họ đang nói về chuyện gì thế?).
- He and his classmates often talk to eachother in English. (Cậu ấy và các bạn cùng lớp thường nói chuyện với nhau bằng tiếng Anh).”
Cách đưa ra lời cảnh báo
Một Số Mẫu Câu Tiếng Anh Giao Tiếp Thông Dụng. 1. Right on! (Great!) - Quá đúng!
2. I did it! (I made it!) - Tôi thành công rồi!
3. Got a minute? - Có rảnh không?
4. About when? - Vào khoảng thời gian nào?
5. I won't take but a minute. - Sẽ không mất nhiều thời gian đâu.
6. Speak up! - Hãy nói lớn lên.
7. Seen Melissa? - Có thấy Melissa không?
8. So we've met again, eh? - Thế là ta lại gặp nhau phải không?
9. Come here. - Đến đây.
10. Come over. - Ghé chơi.
11. Don't go yet. - Đừng đi vội.
12. Please go first. After you. - Xin nhường đi trước. Tôi xin đi sau. 13. Thanks for letting me go first. - Cám ơn đã nhường đường.
14. What a relief. - Thật là nhẹ nhõm.
15. What the hell are you doing? - Anh đang làm cái quái gì thế kia?
13 quy tắc trọng âm trong tiếng anh
Một Số Mẫu Câu Tiếng Anh Giao Tiếp Thông Dụng. 1. Right on! (Great!) - Quá đúng!
2. I did it! (I made it!) - Tôi thành công rồi!
3. Got a minute? - Có rảnh không?
4. About when? - Vào khoảng thời gian nào?
5. I won't take but a minute. - Sẽ không mất nhiều thời gian đâu.
6. Speak up! - Hãy nói lớn lên.
7. Seen Melissa? - Có thấy Melissa không?
8. So we've met again, eh? - Thế là ta lại gặp nhau phải không?
9. Come here. - Đến đây.
10. Come over. - Ghé chơi.
11. Don't go yet. - Đừng đi vội.
12. Please go first. After you. - Xin nhường đi trước. Tôi xin đi sau. 13. Thanks for letting me go first. - Cám ơn đã nhường đường.
14. What a relief. - Thật là nhẹ nhõm.
15. What the hell are you doing? - Anh đang làm cái quái gì thế kia?
Chap2
This chapter introduces the concept of a limit, which is the most important notion in mathematical analysis. The chapter defines what it means for a sequence of numbers to converge or have a limit. Specifically, it provides the precise definition that the limit L of a sequence {an} is the number such that for any positive number ε, there exists an N where if n ≥ N, the distance between an and L is less than ε. The chapter then gives examples of applying this definition to determine the limits of various sequences. It also defines divergence to positive or negative infinity and discusses properties of limits.
Cc10 3 geo_har
The document summarizes geometric and harmonic series. It defines geometric series as having the form ΣCrk where C and r are fixed numbers, and each term is r times the previous term. It provides a theorem showing that a geometric series converges to 1/(1-r) if |r|<1 and diverges if |r|≥1. The harmonic series Σ1/k, where each term is the reciprocal of the term number, is shown to diverge using a grouping argument where the partial sums exceed n/2.
11 ma1aexpandednoteswk3
This document discusses sequences and series of numbers. It begins by introducing sequences and the concept of convergence for sequences. A sequence converges to a limit L if, given any positive number ε, there exists an N such that the terms an of the sequence satisfy |L - an| < ε for all n > N.
It then proves some basic properties of convergence, including that a convergent sequence must be bounded, and that limits are preserved under operations. Cauchy's criterion for convergence is introduced - a sequence is Cauchy if given any ε, there exists an N such that |am - an| < ε for all m, n > N. Every convergent sequence is Cauchy, and C
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Ch11review
- 1. CHAPTER 11 REVIEW QUESTIONS (1) State the − N definition of convergence of a sequence. 1 (2) Prove that limn→∞ 2n+3 = 0 using the above definition of convergence. (3) What does it mean if we say the series an converges? √ n (4) Find the limit of the sequence an = n3 . √ n (5) Does the series n3 converge or diverge? Why? 6 (6) Find the limit of the sequence an = n . 5 6 (7) Does the series converge or diverge? Why? 5n (8) Is it ever possible to conclude a series converges using the nth −term test? (9) Determine whether the following series converge or diverge. Justify your answer. If you use a theorem or test, state the name. 2 a. 3n + 1 2 b. 1 + en nn c. (2n )2 (−1)n d. n2/3 (10) State the definition of absolute convergence and conditional convergence. (11) Is it possible for a series to converge absolutely, but not converge? If yes, give an example. (12) Is it possible for a series to converge, but not converge absolutely? If yes, give an example. (13) Do the following series converge absolutely, conditionally or diverge? (−1)n+1 a. √ 1/3 √ n n cos(nπ) b. √ n2 n 1 c. (n + 1)(n + 2)(n + 3) 1
- 2. 2 CHAPTER 11 REVIEW QUESTIONS (14) State the definition of a power series. (15) For the following state (i) The radius and interval of convergence, (ii) for which values of x does the series converge absolutely, (iii) for which values of x does the series converge conditionally, (iv) for which values of x does the series diverge. ∞ xn a. n=1 n b. (−2)n (n + 1)(x − 1)n ∞ (x + 4)n c. n=1 n3n (16) If the Taylor series generated by f (x) converges, does it necessarly converge to f (x)? (17) (a.) Find the Taylor series generated by f (x) = cos x at a = π/3. (b). Show that the Taylor series converges to cos(x). (18) Find the Taylor series generated by f (x) = 1/x2 at a = 1. (19) Find the Taylor polynomial of order 3: 1 (a.) generated by f (x) = x+2 at a = 0 (b.) generated by f (x) = sin x at a = π/6. (20) State Taylor’s Formula. What is the significance of this formula? (21) State Taylor’s Remainder Estimation Theorem. What is the consequence of this theorem? (22) If cos x is approximated by its Taylor polynomial of order 3 for x such that |x| < 0.5, what is the error estimate? (23) (a.) Express cos(x2 ) as a power series. 1 (b.) Estimate 0 cos(x2 ) with an error of less than 0.001. sin x − x − x3 /6 (24) Use series to evaluate lim . (Do not use L’Hospital’s Rule.) x→0 x5