
# 7.1E: Review of Power Series (Exercises)

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[exer:7.1.1] For each power series use Theorem [thmtype:7.1.3} to find the radius of convergence $$R$$. If $$R>0$$, find the open interval of convergence.

1. $${\displaystyle \sum_{n=0}^\infty {(-1)^n\over2^nn}(x-1)^n}$$
2. $${\displaystyle \sum_{n=0}^\infty 2^nn(x-2)^n}$$
3. $${\displaystyle \sum_{n=0}^\infty {n!\over9^n}x^n}$$
4. $${\displaystyle \sum_{n=0}^\infty{n(n+1)\over16^n}(x-2)^n}$$
5. $${\displaystyle \sum_{n=0}^\infty (-1)^n{7^n\over n!}x^n}$$
6. $${\displaystyle \sum_{n=0}^\infty {3^n\over4^{n+1}(n+1)^2}(x+7)^n}$$

[exer:7.1.2] Suppose there’s an integer $$M$$ such that $$b_m\ne0$$ for $$m\ge M$$, and $\lim_{m\to\infty}\left|b_{m+1}\over b_m\right|=L,\nonumber$ where $$0\le L\le\infty$$. Show that the radius of convergence of $\displaystyle \sum_{m=0}^\infty b_m(x-x_0)^{2m}\nonumber$ is $$R=1/\sqrt L$$, which is interpreted to mean that $$R=0$$ if $$L=\infty$$ or $$R=\infty$$ if $$L=0$$.

[exer:7.1.3] For each power series, use the result of Exercise [exer:7.1.2} to find the radius of convergence $$R$$. If $$R>0$$, find the open interval of convergence.

1. $${\displaystyle \sum_{m=0}^\infty (-1)^m(3m+1)(x-1)^{2m+1}}$$
2. $${\displaystyle \sum_{m=0}^\infty (-1)^m{m(2m+1)\over2^m}(x+2)^{2m}}$$
3. $${\displaystyle \sum_{m=0}^\infty {m!\over(2m)!}(x-1)^{2m}}$$
4. $${\displaystyle \sum_{m=0}^\infty (-1)^m{m!\over9^m}(x+8)^{2m}}$$
5. $${\displaystyle \sum_{m=0}^\infty(-1)^m{(2m-1)\over3^m}x^{2m+1}}$$
6. $${\displaystyle \sum_{m=0}^\infty(x-1)^{2m}}$$

[exer:7.1.4] Suppose there’s an integer $$M$$ such that $$b_m\ne0$$ for $$m\ge M$$, and $\lim_{m\to\infty}\left|b_{m+1}\over b_m\right|=L,\nonumber$ where $$0\le L\le\infty$$. Let $$k$$ be a positive integer. Show that the radius of convergence of $\displaystyle \sum_{m=0}^\infty b_m(x-x_0)^{km}\nonumber$ is $$R=1/\sqrt[k]L$$, which is interpreted to mean that $$R=0$$ if $$L=\infty$$ or $$R=\infty$$ if $$L=0$$.

[exer:7.1.5] For each power series use the result of Exercise [exer:7.1.4} to find the radius of convergence $$R$$. If $$R>0$$, find the open interval of convergence.

1. $${\displaystyle \sum_{m=0}^\infty{(-1)^m\over(27)^m}(x-3)^{3m+2}}$$
2. $${\displaystyle \sum_{m=0}^\infty{x^{7m+6}\over m}}$$
3. $${\displaystyle \sum_{m=0}^\infty{9^m(m+1)\over(m+2)}(x-3)^{4m+2}}$$
4. $${\displaystyle \sum_{m=0}^\infty(-1)^m{2^m\over m!}x^{4m+3}}$$
5. $${\displaystyle \sum_{m=0}^\infty{m!\over(26)^m}(x+1)^{4m+3}}$$
6. $${\displaystyle \sum_{m=0}^\infty{(-1)^m\over8^mm(m+1)}(x-1)^{3m+1}}$$

[exer:7.1.6] Graph $$y=\sin x$$ and the Taylor polynomial $T_{2M+1}(x)=\displaystyle \sum_{n=0}^M{(-1)^nx^{2n+1}\over(2n+1)!}\nonumber$ on the interval $$(-2\pi,2\pi)$$ for $$M=1$$, $$2$$, $$3$$, …, until you find a value of $$M$$ for which there’s no perceptible difference between the two graphs.

[exer:7.1.7] Graph $$y=\cos x$$ and the Taylor polynomial $T_{2M}(x)=\displaystyle \sum_{n=0}^M{(-1)^nx^{2n}\over(2n)!}\nonumber$ on the interval $$(-2\pi,2\pi)$$ for $$M=1$$, $$2$$, $$3$$, …, until you find a value of $$M$$ for which there’s no perceptible difference between the two graphs.

[exer:7.1.8] Graph $$y=1/(1-x)$$ and the Taylor polynomial $T_N(x)=\displaystyle \sum_{n=0}^Nx^n\nonumber$ on the interval $$[0,.95]$$ for $$N=1$$, $$2$$, $$3$$, …, until you find a value of $$N$$ for which there’s no perceptible difference between the two graphs. Choose the scale on the $$y$$-axis so that $$0\le y\le20$$.

[exer:7.1.9] Graph $$y=\cosh x$$ and the Taylor polynomial $T_{2M}(x)=\displaystyle \sum_{n=0}^M{x^{2n}\over(2n)!}\nonumber$ on the interval $$(-5,5)$$ for $$M=1$$, $$2$$, $$3$$, …, until you find a value of $$M$$ for which there’s no perceptible difference between the two graphs. Choose the scale on the $$y$$-axis so that $$0\le y\le75$$.

[exer:7.1.10] Graph $$y=\sinh x$$ and the Taylor polynomial $T_{2M+1}(x)=\displaystyle \sum_{n=0}^M{x^{2n+1}\over(2n+1)!}\nonumber$ on the interval $$(-5,5)$$ for $$M=0$$, $$1$$, $$2$$, …, until you find a value of $$M$$ for which there’s no perceptible difference between the two graphs. Choose the scale on the $$y$$-axis so that $$-75~\le~y\le~75$$.

[exer:7.1.11] $$\vspace*{-5pt}(2+x)y''+xy'+3y$$

[exer:7.1.12] $$(1+3x^2)y''+3x^2y'-2y$$

[exer:7.1.13] $$(1+2x^2)y''+(2-3x)y'+4y$$

[exer:7.1.14] $$(1+x^2)y''+(2-x)y'+3y$$

[exer:7.1.15] $$(1+3x^2)y''-2xy'+4y$$

[exer:7.1.16] Suppose $$y(x)=\displaystyle \sum_{n=0}^\infty a_n(x+1)^n$$ on an open interval that contains $$x_0~=~-1$$. Find a power series in $$x+1$$ for $xy''+(4+2x)y'+(2+x)y.\nonumber$

[exer:7.1.17] Suppose $$y(x)=\displaystyle \sum_{n=0}^\infty a_n(x-2)^n$$ on an open interval that contains $$x_0~=~2$$. Find a power series in $$x-2$$ for $x^2y''+2xy'-3xy.\nonumber$

[exer:7.1.18] Do the following experiment for various choices of real numbers $$a_0$$ and $$a_1$$.

Use differential equations software to solve the initial value problem

$(2-x)y''+2y=0,\quad y(0)=a_0,\quad y'(0)=a_1,\nonumber$

numerically on $$(-1.95,1.95)$$. Choose the most accurate method your software package provides. (See Section 10.1 for a brief discussion of one such method.)

For $$N=2$$, $$3$$, $$4$$, …, compute $$a_2$$, …, $$a_N$$ from Eqn.Equation \ref{eq:7.1.18} and graph $T_N(x)=\displaystyle \sum_{n=0}^N a_nx^n\nonumber$ and the solution obtained in (a) on the same axes. Continue increasing $$N$$ until it is obvious that there’s no point in continuing. (This sounds vague, but you’ll know when to stop.)

[exer:7.1.19] Follow the directions of Exercise [exer:7.1.18} for the initial value problem $(1+x)y''+2(x-1)^2y'+3y=0,\quad y(1)=a_0,\quad y'(1)=a_1,\nonumber$ on the interval $$(0,2)$$. Use Eqns. Equation \ref{eq:7.1.24} and Equation \ref{eq:7.1.25} to compute $$\{a_n\}$$.

[exer:7.1.20] Suppose the series $$\displaystyle \sum_{n=0}^\infty a_nx^n$$ converges on an open interval $$(-R,R)$$, let $$r$$ be an arbitrary real number, and define $y(x)=x^r\displaystyle \sum_{n=0}^\infty a_nx^n=\displaystyle \sum_{n=0}^\infty a_nx^{n+r}\nonumber$ on $$(0,R)$$. Use Theorem [thmtype:7.1.4} and the rule for differentiating the product of two functions to show that \begin{aligned} y'(x)&=&{\displaystyle \sum_{n=0}^\infty (n+r)a_nx^{n+r-1}},\\[10pt] y''(x)&=&{\displaystyle \sum_{n=0}^\infty(n+r)(n+r-1)a_nx^{n+r-2}},\\ &\vdots&\\ y^{(k)}(x)&=&{\displaystyle \sum_{n=0}^\infty(n+r)(n+r-1)\cdots(n+r-k)a_nx^{n+r-k}}\end{aligned}\nonumber on $$(0,R)$$

[exer:7.1.21] $$x^2(1-x)y''+x(4+x)y'+(2-x)y$$

[exer:7.1.22] $$x^2(1+x)y''+x(1+2x)y'-(4+6x)y$$

[exer:7.1.23] $$x^2(1+x)y''-x(1-6x-x^2)y'+(1+6x+x^2)y$$

[exer:7.1.24] $$x^2(1+3x)y''+x(2+12x+x^2)y'+2x(3+x)y$$

[exer:7.1.25] $$x^2(1+2x^2)y''+x(4+2x^2)y'+2(1-x^2)y$$

[exer:7.1.26] $$x^2(2+x^2)y''+2x(5+x^2)y'+2(3-x^2)y$$