# Part 6: Power Series

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## 6.1 Power Series

#### Textbook References

• University Calculus: Early Transcendentals (3rd Ed)
• 9.7

### 6.1.1 Definition

• A power series is a function defined by $$f(x)=\sum_{n=0}^\infty c_nx^n=c_0+c_1x+c_2x^2+c_3x^3+\dots$$ for a coefficient sequence $$\<c_n\>_{n=0}^\infty$$.
• Example Expand the first four terms of the power series $$\sum_{m=0}^\infty (2m+1)x^m$$.
• Example Expand the first four terms of the power series $$\sum_{k=2}^\infty \frac{x^{2k}}{k!}$$.
• The geometric series formula $$\sum_{n=0}^\infty ar^n$$ allows us to simplify certain series where $$|r|<1$$.
• Example Simplify $$f(x)=\sum_{n=0}^\infty x^n=1+x+x^2+x^3+\dots$$ with domain $$|x|<1$$.
• Example Simplify $$p(x)=\sum_{j=1}^\infty (\frac{x}{3})^{2j}= \frac{x^2}{9}+\frac{x^4}{81}+\frac{x^6}{729}+\dots$$ with domain $$|x|<3$$.

### 6.1.2 Domains of Power Series

• The domain of a power series may be determined by applying the Root or Ratio Test to determine for which $$x$$ values the series converges. On the endpoints where these tests are inconclusive, other techniques must be used.
• Example Find the domain of $$f(x)=\sum_{n=1}^\infty\frac{x^n}{n}=x+\frac{x^2}{2}+\frac{x^3}{3}+\dots$$.
• Example Find the domain of $$h(x)=\sum_{n=0}^\infty\frac{(3-2x)^n}{n!}= 1+(3-2x)+\frac{(3-2x)^2}{2}+\frac{(3-2x)^3}{6}+\dots$$.
• Example Find the domain of $$g(x)=\sum_{k=2}^\infty\frac{(3x)^n}{n\ln n}= \frac{9x^2}{2\ln2}+\frac{27x^3}{3\ln3}+\frac{81x^4}{4\ln4}+\dots$$.

### Exercises for 6.1

1. Expand the first four terms of the power series $$\sum_{m=0}^\infty 3^{m+1}x^m$$.
2. Expand the first four terms of the power series $$\sum_{k=1}^\infty \frac{(-x)^k}{k+1}$$.
3. Expand the first four terms of the power series $$\sum_{n=0}^\infty (-1)^n\frac{x^{2n+1}}{(2n+1)!}$$.
4. Simplify $$q(x)=\sum_{n=1}^\infty (1-x)^n=(1-x)+(1-x)^2+(1-x)^3+\dots$$ with domain $$|1-x|<1$$.
5. Simplify $$g(x)=\sum_{j=0}^\infty (2x)^{2j+1}= 2x+8x^3+32x^5+128x^7+\dots$$ with domain $$|x|<\frac{1}{2}$$.
6. Find the domain of $$z(x)=\sum_{n=2}^\infty(-1)^n\frac{x^n}{2n}= \frac{x^2}{4}-\frac{x^3}{6}+\frac{x^4}{8}-\frac{x^5}{10}+\dots$$.
7. Find the domain of $$f(x)=\sum_{i=0}^\infty\frac{(3x)^i}{(2i)!}= 1+\frac{3}{2}x+\frac{9}{24}x^2+\dots$$.
8. Find the domain of $$h(x)=\sum_{k=0}^\infty\frac{(x-2)^k}{k^2+1}= 1+\frac{x-2}{2}+\frac{(x-2)^2}{5}+\frac{(x-2)^3}{10}+\dots$$.
9. (OPTIONAL) Find the domain of $$g(x)=\sum_{m=3}^\infty(\frac{1}{m}-\frac{1}{m+1})x^m$$.
10. (QUIZ) Expand the first four terms of the power series $$\sum_{k=0}^\infty \frac{(-x)^{2k+1}}{(2k)!}$$.
• $$1+x^2+\frac{x^3}{8}+\frac{x^5}{15}+\dots$$
• $$-x-\frac{x^3}{2}-\frac{x^5}{24}-\frac{x^7}{720}-\dots$$
• $$x-\frac{x^3}{3}+\frac{x^6}{18}-\frac{x^{10}}{27}+\dots$$
11. (QUIZ) Simplify $$f(x)=\sum_{n=1}^\infty (-x)^{n-1}=1-x+x^2-x^3+\dots$$ with domain $$|x|<1$$.
• $$f(x)=\frac{1}{1+x}$$
• $$f(x)=\frac{2x}{1-x}$$
• $$f(x)=\frac{1}{x}+2$$
12. (QUIZ) Find the domain of $$f(x)=\sum_{m=2}^\infty\frac{(-2x)^m}{m}= \frac{4x^2}{2}-\frac{8x^3}{3}+\frac{16x^4}{4}-\frac{32x^5}{5}+\dots$$.
• $$-\frac{1}{2}<x\leq\frac{1}{2}$$
• $$-1<x<1$$
• $$0\leq x<2$$

Solutions

## 6.2 Taylor and Maclaurin Series

• University Calculus: Early Transcendentals (3rd Ed)
• 9.8

### 6.2.1 Power Series from Functions

• Let $$f(x)$$ have derivatives of all orders nearby $$a$$. Then the Taylor series generated by $$f$$ at $$a$$ is given by $$\sum_{k=0}^\infty\frac{f^{(k)}(a)}{k!}(x-a)^k$$, where $$f^{(k)}(a)$$ is the $$k^{th}$$ derivative of $$f$$ at $$a$$.
• A Maclaurin series is a Taylor series where $$a=0$$.
• A Taylor/Maclaurin series is said to converge to its generating function if it is equal to it for all members of its domain.
• Example Let $$f(x)=\frac{1}{1+x}$$ with the domain $$-1<x<1$$, and guess a formula for $$f^{(k)}(0)$$ by computing its first few terms. Then show that the Maclaurin series generated by $$f$$ converges to $$f$$.
• Example Let $$g(x)=\frac{2}{x}$$ with the domain $$0<x<4$$, and guess a formula for $$g^{(k)}(2)$$ by computing its first few terms. Then show that the Taylor series generated by $$g$$ at $$2$$ converges to $$g$$.
• It can be shown that $$f$$ defined by $$f(0)=0$$ and $$f(x)=e^{-1/x^2}$$ otherwise satisfies $$f^{(k)}(0)=0$$, giving an example of a function which doesn’t converge to its Taylor series.

### 6.2.2 Maclaurin Series for $$e^x$$, $$\sin x$$, $$\cos x$$

• The following Maclaurin Series can be shown to converge to their generating functions:
• $$e^x=\sum_{k=0}^\infty\frac{x^k}{k!} = 1+x+\frac{x^2}{2}+\frac{x^3}{6}+\dots$$
• $$\cos x = \sum_{k=0}^\infty(-1)^k\frac{x^{2k}}{(2k)!} = 1-\frac{x^2}{2}+\frac{x^4}{24}-\frac{x^6}{720}+\dots$$
• $$\sin x = \sum_{k=0}^\infty(-1)^k\frac{x^{2k+1}}{(2k+1)!} = x-\frac{x^3}{6}+\frac{x^5}{120}-\frac{x^7}{5040}+\dots$$
• Example Show how to generate the Maclaurin series for $$e^x$$.
• Example Show how to generate the Maclaurin series for $$\sin x$$.

### 6.2.3 Composition, Differentiation, and Integration of Power Series

• Power series may be manipulated by the multiplication and composition of continuous functions.
• Example Find a power series converging to $$x\cos(-x^2)$$.
• Power series may be differentiated and integrated term-by-term.
• Example Find a power series converging to $$\frac{1}{x^2-2x+1}$$ for $$-1<x<1$$.
• Example Find a power series converging to $$\tan^{\leftarrow}(x)$$ for $$-1<x<1$$.

### Exercises for 6.2

1. Let $$f(x)=\frac{1}{1-x}$$ with the domain $$-1<x<1$$, and guess a formula for $$f^{(k)}(0)$$ by computing its first few terms. Then show that the Maclaurin series generated by $$f$$ converges to $$f$$.
2. Let $$g(x)=\frac{3}{x}$$ with the domain $$0<x<6$$, and guess a formula for $$g^{(k)}(3)$$ by computing its first few terms. Then show that the Taylor series generated by $$g$$ at $$3$$ converges to $$g$$.
3. Show how to generate the Maclaurin series $$\sum_{k=0}^\infty(-1)^k\frac{x^{2k}}{(2k)!}$$ for $$\cos x$$.
4. Find the Maclaurin series for $$\sinh x$$.
5. Find a power series converging to $$\frac{x^3}{e^{x^2}}$$.
6. Find a power series converging to $$\frac{1}{x^2+2x+1}$$ for $$|x|<1$$. (Hint: begin with the power series for $$\frac{1}{1+x}$$ and then differentiate term-by-term.)
7. Find a power series converging to $$\ln|x|$$ for $$0<x<2$$. (Hint: begin with the power series for $$\frac{1}{1+x}$$ and then integrate term-by-term.)
8. (QUIZ) Generate the Maclaurin Series for $$\cosh x$$.
• $$\sum_{k=0}^\infty(-1)^k\frac{x^{2k+1}}{(2k+1)!}$$
• $$\sum_{k=0}^\infty\frac{x^{2k}}{(2k)!}$$
• $$\sum_{k=0}^\infty\frac{x^k}{(k+1)!}$$
9. (QUIZ) Find a power series converging to $$x\cos x$$.
• $$\sum_{k=0}^\infty(-1)^k\frac{x^{2k+1}}{(2k)!}$$
• $$\sum_{k=0}^\infty(-1)^k\frac{x^{2k}}{(2k+2)!}$$
• $$\sum_{k=0}^\infty\frac{x^{2k+2}}{(2k+1)!}$$
10. (QUIZ) Find a power series converging to $$\frac{d}{dx}[\sin(x^2)]$$.
• $$\sum_{k=0}^\infty\frac{(-x^2)^{4k+3}}{(4k+2)(2k)!}$$
• $$\sum_{k=0}^\infty\frac{(2k+1)x^{2k+2}}{(2k+1)!}$$
• $$\sum_{k=0}^\infty(-1)^k\frac{(4k+2)x^{4k+1}}{(2k+1)!}$$

Solutions 1-4

Solutions 5-10

## 6.3 Convergence of Taylor Series

• University Calculus: Early Transcendentals (3rd Ed)
• 9.9

### 6.3.1 Taylor’s Formula

• Taylor’s Formula guarantees that if $$f$$ has derivatives of all orders on an open interval containing $$a$$, then for every nonnegative integer $$n$$ and $$x$$ in that same interval, $$f(x)=\left(\sum_{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a)^k\right)+R_n(x)$$ where the error term is given by $$R_n(x)=\frac{f^{(n+1)}(x_n)}{(n+1)!}(x-a)^{n+1}$$ for some number $$x_n$$ between $$a$$ and $$x$$.
• Example Use the fact that $$e<4$$ and Taylor’s Formula to estimate the value of $$\sqrt{e}$$ with an error no greater than $$0.01$$.
• Example Use Taylor’s Formula to estimate the value of $$\sin(0.1)$$ with an error no greater than $$0.0001$$.

### 6.3.2 Convergence of Taylor and Maclaurin Series

• A Taylor series converges to its generating function when $$\lim_{n\to\infty}|R_n(x)|=0$$.
• Example Prove that $$e^x=\sum_{k=0}^\infty\frac{x^k}{k!}$$.
• Example Prove that $$\cos(x)=\sum_{k=0}^\infty(-1)^k\frac{x^{2k}}{(2k)!}$$.

### Exercises for 6.3

1. Use the fact that $$e<3$$ and Taylor’s Formula to estimate the value of $$e$$ with an error no greater than $$0.001$$.
2. Use Taylor’s Formula to estimate the value of $$\cos(0.1)$$ with an error no greater than $$0.0001$$.
3. Use Taylor’s Formula to estimate the value of $$\sin(1)$$ with an error no greater than $$0.01$$.
4. Prove that $$\sin(x)=\sum_{k=0}^\infty(-1)^k\frac{x^{2k+1}}{(2k+1)!}$$.
5. Use the fact that $$|\sinh(x_n)|\leq|\cosh(x_n)|\leq\cosh(x)$$ for any $$x_n$$ between $$0$$ and $$x$$ to prove that $$\cosh(x)=\sum_{k=0}^\infty\frac{x^{2k}}{(2k)!}$$.
6. Reprove $$\cosh(x)=\sum_{k=0}^\infty\frac{x^{2k}}{(2k)!}$$ by using its definition $$\cosh(x)=\frac{1}{2}(e^x+e^{-x})$$ along with the Maclaurin series for $$e^x$$.
7. (OPTIONAL) Prove that $$|\sinh(x_n)|\leq|\cosh(x_n)|\leq\cosh(x)$$ for any $$x_n$$ between $$0$$ and $$x$$.
8. (QUIZ) Which of these Maclaurin Series is most appropriate for approximating $$e^{-1/2}=\frac{1}{\sqrt e}$$?
• $$\sum_{k=0}^\infty(-1)^k\frac{x^{2k+1}}{(2k+1)!}$$
• $$\sum_{k=0}^\infty(-1)^k\frac{x^{2k}}{(2k)!}$$
• $$\sum_{k=0}^\infty\frac{x^k}{k!}$$
9. (QUIZ) Find the error term $$R_n(x)$$ from the Taylor Formula for $$e^x$$, where $$x_n$$ is between $$0$$ and $$x$$.
• $$R_n(x)=\frac{x^{n+1}}{n!}$$
• $$R_n(x)=\frac{e^{x_n}}{(n+1)!}x^{n+1}$$
• $$R_n(x)=\frac{1}{e^{x_n/2}(n+1)!}x^n$$
10. (QUIZ) Use Taylor’s Formula to approximate $$e^{-1/2}=\frac{1}{\sqrt e}$$ with an error no greater than $$\frac{1}{1000}=0.001$$. (Hint: $$-1/2\leq x_n\leq 0$$.)
• $$\frac{1}{\sqrt e}\approx 0.604$$
• $$\frac{1}{\sqrt e}\approx 0.607$$
• $$\frac{1}{\sqrt e}\approx 0.609$$

Solutions 1-7

Solutions 8-10