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# 3.1E: Exercises

• • Contributed by William F. Trench
• Andrew G. Cowles Distinguished Professor Emeritus (Mathamatics) at Trinity University
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## Exercise $$\PageIndex{1}$$

For each power series use Theorem $$(3.1.3)$$ to find the radius of convergence $$R$$. If $$R>0$$, find the open interval of convergence.

(a) $$\displaystyle{\sum_{n=0}^\infty {(-1)^n\over2^nn}(x-1)^n}$$

(b) $$\displaystyle{\sum_{n=0}^\infty 2^nn(x-2)^n}$$

(c) $$\displaystyle{\sum_{n=0}^\infty {n!\over9^n}x^n}$$

(d) $$\displaystyle{\sum_{n=0}^\infty{n(n+1)\over16^n}(x-2)^n}$$

(e) $$\displaystyle{\sum_{n=0}^\infty (-1)^n{7^n\over n!}x^n}$$

(f) $$\displaystyle{\sum_{n=0}^\infty {3^n\over4^{n+1}(n+1)^2}(x+7)^n}$$

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## Exercise $$\PageIndex{2}$$

Suppose there's an integer $$M$$ such that $$b_m\ne0$$ for $$m\ge M$$, and

\begin{eqnarray*}
\lim_{m\to\infty}\left|b_{m+1}\over b_m\right|=L,
\end{eqnarray*}

where $$0\le L\le\infty$$. Show that the radius of convergence of

\begin{eqnarray*}
\sum_{m=0}^\infty b_m(x-x_0)^{2m}
\end{eqnarray*}

is $$R=1/\sqrt L$$, which is interpreted to mean that $$R=0$$ if $$L=\infty$$ or $$R=\infty$$ if $$L=0$$.

Hint: Apply Theorem $$(3.1.3)$$ to the series $$\sum_{m=0}^\infty b_mz^m$$ and then let $$z=(x-x_0)^2$$.

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## Exercise $$\PageIndex{3}$$

For each power series, use the result of Exercise $$(3.1E.2)$$ to find the radius of convergence $$R$$. If $$R>0$$, find the open interval of convergence.

(a) $$\displaystyle{\sum_{m=0}^\infty (-1)^m(3m+1)(x-1)^{2m+1}}$$

(b) $$\displaystyle{\sum_{m=0}^\infty (-1)^m{m(2m+1)\over2^m}(x+2)^{2m}}$$

(c) $$\displaystyle{\sum_{m=0}^\infty {m!\over(2m)!}(x-1)^{2m}}$$

(d) $$\displaystyle{\sum_{m=0}^\infty (-1)^m{m!\over9^m}(x+8)^{2m}}$$

(e) $$\displaystyle{\sum_{m=0}^\infty(-1)^m{(2m-1)\over3^m}x^{2m+1}}$$

(f) $$\displaystyle{\sum_{m=0}^\infty(x-1)^{2m}}$$

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## Exercise $$\PageIndex{4}$$

Suppose there's an integer $$M$$ such that $$b_m\ne0$$ for \)m\ge M\), and

\begin{eqnarray*}
\lim_{m\to\infty}\left|b_{m+1}\over b_m\right|=L,
\end{eqnarray*}

where $$0\le L\le\infty$$. Let $$k$$ be a positive integer. Show that the radius of convergence of

\begin{eqnarray*}
\sum_{m=0}^\infty b_m(x-x_0)^{km}
\end{eqnarray*}

is $$R=1/\sqrt[k]L$$, which is interpreted to mean that $$R=0$$ if $$L=\infty$$ or $$R=\infty$$ if $$L=0$$.

Hint: Apply Theorem $$(3.1.3)$$ to the series $$\sum_{m=0}^\infty b_mz^m$$ and then let $$z=(x-x_0)^k$$.

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## Exercise $$\PageIndex{5}$$

For each power series use the result of Exercise $$(3.1E.4)$$ to find the radius of convergence $$R$$. If $$R>0$$, find the open interval of convergence.

(a) $$\displaystyle{\sum_{m=0}^\infty{(-1)^m\over(27)^m}(x-3)^{3m+2}}$$

(b) $$\displaystyle{\sum_{m=0}^\infty{x^{7m+6}\over m}}$$

(c) $$\displaystyle{\sum_{m=0}^\infty{9^m(m+1)\over(m+2)}(x-3)^{4m+2}}$$

(d) $$\displaystyle{\sum_{m=0}^\infty(-1)^m{2^m\over m!}x^{4m+3}}$$

(e) $$\displaystyle{\sum_{m=0}^\infty{m!\over(26)^m}(x+1)^{4m+3}}$$

(f) $$\displaystyle{\sum_{m=0}^\infty{(-1)^m\over8^mm(m+1)}(x-1)^{3m+1}}$$

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## Exercise $$\PageIndex{6}$$

Graph $$y=\sin x$$ and the Taylor polynomial

\begin{eqnarray*}
T_{2M+1}(x)=\sum_{n=0}^M{(-1)^nx^{2n+1}\over(2n+1)!}
\end{eqnarray*}

on the interval $$(-2\pi,2\pi)$$ for $$M=1$$, $$2$$, $$3$$, $$\dots$$, until you find a value of $$M$$ for which there's no perceptible difference between the two graphs.

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## Exercise $$\PageIndex{7}$$

Graph $$y=\cos x$$ and the Taylor polynomial

\begin{eqnarray*}
T_{2M}(x)=\sum_{n=0}^M{(-1)^nx^{2n}\over(2n)!}
\end{eqnarray*}

on the interval $$(-2\pi,2\pi)$$ for $$M=1$$, $$2$$, $$3$$, $$\dots$$, until you find a value of $$M$$ for which there's no perceptible difference between the two graphs.

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## Exercise $$\PageIndex{8}$$

Graph $$y=1/(1-x)$$ and the Taylor polynomial

\begin{eqnarray*}
T_N(x)=\sum_{n=0}^Nx^n
\end{eqnarray*}

on the interval $$[0,.95]$$ for $$N=1$$, $$2$$, $$3$$, $$\dots$$, 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 $$0\le y\le20$$.

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## Exercise $$\PageIndex{9}$$

Graph $$y=\cosh x$$ and the Taylor polynomial

\begin{eqnarray*}
T_{2M}(x)=\sum_{n=0}^M{x^{2n}\over(2n)!}
\end{eqnarray*}

on the interval $$(-5,5)$$ for $$M=1$$, $$2$$, $$3$$, $$\dots$$, 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 $$0\le y\le75$$.

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## Exercise $$\PageIndex{10}$$

Graph $$y=\sinh x$$ and the Taylor polynomial

\begin{eqnarray*}
T_{2M+1}(x)=\sum_{n=0}^M{x^{2n+1}\over(2n+1)!}
\end{eqnarray*}

on the interval $$(-5,5)$$ for $$M=0$$, $$1$$, $$2$$, $$\dots$$, 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 $$-75~\le~y\le~75$$.

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In Exercises $$(3.1E.11)$$ to $$(3.1E.15)$$, find a power series solution $$y(x)=\sum_{n=0}^\infty a_nx^n$$.

## Exercise $$\PageIndex{11}$$

$$(2+x)y''+xy'+3y$$

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## Exercise $$\PageIndex{12}$$

$$(1+3x^2)y''+3x^2y'-2y$$

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## Exercise $$\PageIndex{13}$$

$$(1+2x^2)y''+(2-3x)y'+4y$$

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## Exercise $$\PageIndex{14}$$

$$(1+x^2)y''+(2-x)y'+3y$$

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## Exercise $$\PageIndex{15}$$

$$(1+3x^2)y''-2xy'+4y$$

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## Exercise $$\PageIndex{16}$$

Suppose $$y(x)=\sum_{n=0}^\infty a_n(x+1)^n$$ on an open interval that contains $$x_0=-1$$. Find a power series in $$x+1$$ for

\begin{eqnarray*}
xy''+(4+2x)y'+(2+x)y.
\end{eqnarray*}

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## Exercise $$\PageIndex{17}$$

Suppose $$y(x)=\sum_{n=0}^\infty a_n(x-2)^n$$ on an open interval that contains $$x_0=2$$. Find a power series in $$x-2$$ for

\begin{eqnarray*}
x^2y''+2xy'-3xy.
\end{eqnarray*}

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## Exercise $$\PageIndex{18}$$

Do the following experiment for various choices of real numbers $$a_0$$ and $$a_1$$.

(a) Use differential equations software to solve the initial value problem

\begin{eqnarray*}
\end{eqnarray*}

numerically on $$(-1.95,1.95)$$. Choose the most accurate method your software package provides. (See Section 3.1 for a brief discussion of one such method.)

(b) For $$N=2$$, $$3$$, $$4$$, $$\dots$$, compute $$a_2$$, $$\dots$$, $$a_N$$ from Equation $$(3.1.18)$$ and graph

\begin{eqnarray*}
T_N(x)=\sum_{n=0}^N a_nx^n
\end{eqnarray*}

and the solution obtained in part (a) on the same axes. Continue increasing $$N$$ until it's obvious that there's no point in continuing. (This sounds vague, but you'll know when to stop.)

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## Exercise $$\PageIndex{19}$$

Follow the directions of Exercise $$(3.1.18)$$ for the initial value problem

\begin{eqnarray*}
\end{eqnarray*}

on the interval $$(0,2)$$. Use Equations $$(3.1.24) \mbox{ and } (3.1.25)$$ to compute $$\{a_n\}$$.

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## Exercise $$\PageIndex{20}$$

Suppose the series $$\sum_{n=0}^\infty a_nx^n$$ converges on an open interval $$(-R,R)$$, let $$r$$ be an arbitrary real number, and define

\begin{eqnarray*}
y(x)=x^r\sum_{n=0}^\infty a_nx^n=\sum_{n=0}^\infty a_nx^{n+r}
\end{eqnarray*}

on $$(0,R)$$. Use Theorem $$(3.1.4)$$ and the rule for differentiating the product of two functions to show that

\begin{eqnarray*}
y'(x)&=&\displaystyle{\sum_{n=0}^\infty (n+r)a_nx^{n+r-1}},\\
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{eqnarray*}

on $$(0,R)$$

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In Exercises $$(3.1E.21)$$ to $$(3.1E.26)$$, let $$y$$ be as defined in Exercise $$(3.1E.20)$$, and write the given expression in the form $$x^r\sum_{n=0}^\infty b_nx^n$$.

## Exercise $$\PageIndex{21}$$

$$x^2(1-x)y''+x(4+x)y'+(2-x)y$$

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## Exercise $$\PageIndex{22}$$

$$x^2(1+x)y''+x(1+2x)y'-(4+6x)y$$

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## Exercise $$\PageIndex{23}$$

$$x^2(1+x)y''-x(1-6x-x^2)y'+(1+6x+x^2)y$$

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## Exercise $$\PageIndex{24}$$

$$x^2(1+3x)y''+x(2+12x+x^2)y'+2x(3+x)y$$

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## Exercise $$\PageIndex{25}$$

$$x^2(1+2x^2)y''+x(4+2x^2)y'+2(1-x^2)y$$

## Exercise $$\PageIndex{26}$$
$$x^2(2+x^2)y''+2x(5+x^2)y'+2(3-x^2)y$$