Chapter 8: Infinite Sequences and Series

Section 8.4: Power Series

Example 8.4.18

$$ Determine the radius of convergence and the interval of convergence for the power series $\sum _{n\=0}^{\infty }\frac{2^n}{n\!}{\left(3 x\+5\right)}^n$.   Even though (7) in Table 8.4.1 claims that absolute convergence at one end of the interval of convergence implies absolute convergence at the other, if the convergence at an endpoint is absolute, verify that it also absolute at the other. $$ Solution Mathematical Solution $$ •  Since the given power series contains the powers ${\left(3 x\+5\right)}^n$, the radius of convergence is given by   $R\=\frac{1}{3}\underset{n\to \infty }{lim}{a}_n\/a_{n\+1}$ = $\frac{1}{3}\underset{n\to \infty }{lim}\frac{2^n\/n\!}{2^{n\+1}\/\left(n\+1\right)\!}\=\frac{1}{3}\underset{n\to \infty }{lim}\frac{n\+1}{2}\=\infty$     •  Hence, the interval of convergence is $\left(-\infty \,\infty \right)$. $$ Maple Solution $$ Define the general coefficient $a_n$ as a function of $n$  •  Write $a\left(n\right)\=\dots$ Context Panel: Assign Function $a\left(n\right)\=2^n\/n\!$$\stackrel{\text{assign as function}}{\to }$$a$ Obtain the radius of convergence •  Calculus palette: Limit template Context Panel: Assign Name $R\=\frac{1}{3} \underset{n\to \infty }{lim}\frac{a\left(n\right)}{a\left(n\+1\right)}$$\stackrel{\text{assign}}{\to }$ Display $R$, the radius of convergence •  Write $R$ Context Panel: Evaluate and Display Inline $R$ = $\mathrm{\∞}$$$ $$ •  Hence, the interval of convergence is $\left(-\infty \,\infty \right)$. •  Maple sums this series to $\sum _{n\=0}^{\infty }\frac{2^n}{n\!} {\left(3 x\+5\right)}^n$ = ${\ⅇ}^{6x\+10}$. $$$$

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