# Power Series Stewart Calculus Homework

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## Calculus 1B (001 LEC) Fall 2014

Instructor: Alexander Paulin.
e-mail: apaulinberkeley.edu.
Office: 796 Evans Hall.
Office hours : Monday, Wednesday and Friday 2-4pm. Tuesday and Thursday, 1pm-2pm.
Lectures: Mondays, Wednesdays and Fridays, 8am-9am.155 Dwinelle Hall. There will be no lectures on Septmenber 1 or November 28.
Discussion sections: Mondays, Wednesday and Fridays, at various times (see below).
Enrollment: For question about enrollment contact Thomas Brown.

### Announcements

• (9/1) On the week of Thanksgiving the homework submission/quiz will be on Monday as opposed to Wednesday. I will therefore post the homework on monday the previous week.
• (8/29) I've posted my notes from the first lecture below. Just click on the link directly from the lecture schedule. I'll try to post my notes from each week on Friday evenings.
• (8/27) There will be no official homework due on September 3. However, as a warm up, I'm giving out a series of exercises (homework 0, I suppose) from the first half of the textbook, i.e. material covered in Calculus 1A. I strongly recommend you do these. It's easy to forget the basics after such a long break and it's a great way to get back in the swing of things.
• (8/27) Make sure to read the course policy and the detailed syllabus.
• ### Textbook

The textbook for this course is: Stewart, Single Variable Calculus: Early Transcendentals for UC Berkeley, 7th edition (ISBN: 978-0538498678, Cengage).

This is a custom edition containing chapters 1-9, 11 and 17 of Stewart's "Calculus: Early Transcendentals", 7th edition. The regular edition is also fine, it just contains extra chapters covered in math 53. The 6th edition is also acceptable, but you will need to watch for differences in the numbering of assigned homework problems. In chronological order, we'll cover the material in chapters 7, 8, 11, 9 and 17.

Weekly homework and quizzes 20%; two midterms 20% each; final exam 40%. The lowest midterm can be dropped and replaced by the final exam grade. There will be no make-up exams. This grading policy allows you to miss one midterm, but check your schedule to make sure you have no conflict for the final exam.

Make sure to read the detailed course policy for important information.

### Homework

Homework assignments are due each Wednesday in section; they will be posted here. The first homework will be due on September 10. For more detailed information see the course policy.

Homework 0 and Solutions 0

Homework 1 and Solutions 1

Homework 2 and Solutions 2

Homework 3 and Solutions 3

Homework 4 and Solutions 4

Homework 5 and Solutions 5

Homework 6 and Solutions 6

Homework 7 and Solutions 7

Homework 8 and Solutions 8

Homework 9 and Solutions 9

Homework 10 and Solutions 10

Homework 11 and Solutions 11

### Exams

There will be two midterms, the first on Monday September 29 and the second on Friday October 31, both from 8am to 9am in the usual lecture room. The final exam will be on Monday December 15 (7-10pm).

Practice First Midterm: Good Solutions and Bad Solutions

Here is the first midterm, together with solutions. A rough breakdown of the letter grades for the first midterm is as follows:

 Score Grade 85-100 A 65-84 B 45-64 C 35-44 D

The mean score was 71 and the median was 73. Keep in mind that these letter grades are estimates only - only the numbers are used to compute your final grade.

Practice Second Midterm and solutions.

Here is the second midterm, together with solutions.A rough breakdown of the letter grades for the second midterm is as follows:

 Score Grade 81-100 A 61-80 B 41-60 C 30-40 D

The mean score was 64 and the median was 65. Keep in mind that these letter grades are estimates only - only the numbers are used to compute your final grade.

Practice Final Exam and solutions.

Here is a link to many past exams. Of these, the ones I recommend trying first are Ribet97, Sarason96 and Reshetikhin03. Remember though, when you look at these older exams the instructor may have focussed on different things and so they may be quite different from mine.

### Syllabus

In Math 1A or elsewhere, you studied functions of a single variable, limits, and continuity. You learned about derivatives, which describe how functions change, and which can be used to help find maxima and minima of functions. You also learned about integrals which describe the aggregate behavior of a function over an interval, such as the area under a curve or the average of a varying quantity. The derivative and the integral are tied together in the fundamental theorem of calculus, one version of which relates the integral of the derivative of a function over an interval to the values of the function at the endpoints of the interval.

In this course we will continue the study of calculus in three parts as follows:

• The first part of the course is about techniques of integration (sections 7.1 to 7.8 of the book). As you should already know, differentiation is relatively straightforward: if you know the derivatives of elementary functions, and rules such as the product rule and the chain rule, then you can differentiate just about any function you will ever come across. Integration, on the other hand, is hard. Sometimes it is even impossible to integrate a given function explicitly in terms of known functions. We will introduce a collection of useful tricks with which you can integrate many functions. The hard part is to figure out which trick(s) to use in a given situation. For integrals which we cannot evaluate explicitly, we will learn how to find good approximations to the answer.
• The second part of the course is about sequences and series (chapter 11 of the book). This can be regarded as the general theory of approximating things. This part of the course is subtle and involves new ways of thinking. It may be a lot harder than the first part, especially if you have seen some of the first part before.
• The third part of the course is an introduction to ordinary differential equations (chapters 9 and 17 of the book). Here one tries to understand a function, given an equation involving the function and its derivatives. ("Ordinary" means that we consider functions of a single variable. Functions of several variables enter into "partial" differential equations, which you can learn about in a more advanced course.) The theory of differential equations is perhaps the most interesting part of calculus, is the subject of much present-day research, and has many real-world applications. Our study of differential equations will make use of most of the calculus we have done so far.

Here is the lecture schedule for the course:

### Discussion sections

 Section Time Room Instructor e-mail Office hours 101 MWF 9-10am 385 Leconte Shen, Chen 102 MWF 9-10am 122 Wheeler Wu, Kumming 103 MWF 9-10am 200 Wheeler Xiao, Jianwei 104 MWF 10-11am 210 Wheeler Xiao, Jianwei 105 MWF 11-12pm 6 Evans Shen, Chen 106 MWF 11-12pm 30 Wheeler Hollowood, D 107 MWF 12-1pm B51 Hildebrand Fei, Yang 108 MWF 12-1pm 6 Evans Hollowood, D 109 MWF 1-2pm 5 Evans Fernando, R 110 MWF 2-3pm 75 Evans Fernando, R 111 MWF 3-4pm 81 Evans Leake, Jonathan 112 MWF 4-5pm 179 Stanley Fei, Yang 113 MWF 4-5pm 55 Evans Eng, Emily 114 MWF 5-6pm 51 Evans Eng, Emily 115 MWF 1-3pm 230C Stephens Rusciano, Alex 116 MWF 2-3pm 61 Evans Leake, Jonathan 117 MWF 10-11am 5 Evans Wu, Kumming

### Resources

The Student Learning Center provides support for this class, including study groups, review sessions for exams, and drop-in tutoring.

Recall that we were able to analyze all geometric series "simultaneously'' to discover that $$\sum_{n=0}^\infty kx^n = {k\over 1-x},$$ if $|x|< 1$, and that the series diverges when $|x|\ge 1$. At the time, we thought of $x$ as an unspecified constant, but we could just as well think of it as a variable, in which case the series $$\sum_{n=0}^\infty kx^n$$ is a function, namely, the function $k/(1-x)$, as long as $|x|< 1$. While $k/(1-x)$ is a reasonably easy function to deal with, the more complicated $\sum kx^n$ does have its attractions: it appears to be an infinite version of one of the simplest function types—a polynomial. This leads naturally to the questions: Do other functions have representations as series? Is there an advantage to viewing them in this way?

The geometric series has a special feature that makes it unlike a typical polynomial—the coefficients of the powers of $x$ are the same, namely $k$. We will need to allow more general coefficients if we are to get anything other than the geometric series.

Definition 11.8.1 A power series has the form $$\ds\sum_{n=0}^\infty a_nx^n,$$ with the understanding that $\ds a_n$ may depend on $n$ but not on $x$.

Example 11.8.2 $\ds\sum_{n=1}^\infty {x^n\over n}$ is a power series. We can investigate convergence using the ratio test: $$\lim_{n\to\infty} {|x|^{n+1}\over n+1}{n\over |x|^n} =\lim_{n\to\infty} |x|{n\over n+1} =|x|.$$ Thus when $|x|< 1$ the series converges and when $|x|>1$ it diverges, leaving only two values in doubt. When $x=1$ the series is the harmonic series and diverges; when $x=-1$ it is the alternating harmonic series (actually the negative of the usual alternating harmonic series) and converges. Thus, we may think of $\ds\sum_{n=1}^\infty {x^n\over n}$ as a function from the interval $[-1,1)$ to the real numbers.

A bit of thought reveals that the ratio test applied to a power series will always have the same nice form. In general, we will compute $$\lim_{n\to\infty} {|a_{n+1}||x|^{n+1}\over |a_n||x|^n} =\lim_{n\to\infty} |x|{|a_{n+1}|\over |a_n|} = |x|\lim_{n\to\infty} {|a_{n+1}|\over |a_n|} =L|x|,$$ assuming that $\ds \lim |a_{n+1}|/|a_n|$ exists. Then the series converges if $L|x|< 1$, that is, if $|x|< 1/L$, and diverges if $|x|>1/L$. Only the two values $x=\pm1/L$ require further investigation. Thus the series will definitely define a function on the interval $(-1/L,1/L)$, and perhaps will extend to one or both endpoints as well. Two special cases deserve mention: if $L=0$ the limit is $0$ no matter what value $x$ takes, so the series converges for all $x$ and the function is defined for all real numbers. If $L=\infty$, then no matter what value $x$ takes the limit is infinite and the series converges only when $x=0$. The value $1/L$ is called the radius of convergence of the series, and the interval on which the series converges is the interval of convergence .

Consider again the geometric series, $$\sum_{n=0}^\infty x^n={1\over 1-x}.$$ Whatever benefits there might be in using the series form of this function are only available to us when $x$ is between $-1$ and $1$. Frequently we can address this shortcoming by modifying the power series slightly. Consider this series: $$\sum_{n=0}^\infty {(x+2)^n\over 3^n}= \sum_{n=0}^\infty \left({x+2\over 3}\right)^n={1\over 1-{x+2\over 3}}= {3\over 1-x},$$ because this is just a geometric series with $x$ replaced by $(x+2)/3$. Multiplying both sides by $1/3$ gives $$\sum_{n=0}^\infty {(x+2)^n\over 3^{n+1}}={1\over 1-x},$$ the same function as before. For what values of $x$ does this series converge? Since it is a geometric series, we know that it converges when \eqalign{ |x+2|/3&< 1\cr |x+2|&< 3\cr -3 < x+2 &< 3\cr -5< x&< 1.\cr } So we have a series representation for $1/(1-x)$ that works on a larger interval than before, at the expense of a somewhat more complicated series. The endpoints of the interval of convergence now are $-5$ and $1$, but note that they can be more compactly described as $-2\pm3$. We say that $3$ is the radius of convergence, and we now say that the series is centered at $-2$.

Definition 11.8.3 A power series centered at $a$ has the form $$\ds\sum_{n=0}^\infty a_n(x-a)^n,$$ with the understanding that $\ds a_n$ may depend on $n$ but not on $x$.

## Exercises 11.8

Find the radius and interval of convergence for each series. In exercises 3 and 4, do not attempt to determine whether the endpoints are in the interval of convergence.

Ex 11.8.1 $\ds\sum_{n=0}^\infty n x^n$ (answer)

Ex 11.8.2 $\ds\sum_{n=0}^\infty {x^n\over n!}$ (answer)

Ex 11.8.3 $\ds\sum_{n=1}^\infty {n!\over n^n}x^n$ (answer)

Ex 11.8.4 $\ds\sum_{n=1}^\infty {n!\over n^n}(x-2)^n$ (answer)

Ex 11.8.5 $\ds\sum_{n=1}^\infty {(n!)^2\over n^n}(x-2)^n$ (answer)

Ex 11.8.6 $\ds\sum_{n=1}^\infty {(x+5)^n\over n(n+1)}$ (answer)

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