Question

Find the intervals of convergence for the function. f(x)=sum_(n=1)^infinity x**n/n**2 32324243232 incorrect: your answer is incorrect. (b) find the intervals of convergence for f '.

Answer 1

$\displaystyle\sum_{n\ge1}\frac{x^n}{n^2}$

will converge as long as

$\displaystyle\lim_{n\to\infty}\left|\frac{\frac{x^{n+1}}{(n+1)^2}}{\frac{x^n}{n^2}}\right|<1$

according to the ratio test. We have

$\displaystyle\lim_{n\to\infty}\left|\frac{\frac{x^{n+1}}{(n+1)^2}}{\frac{x^n}{n^2}}\right|=|x|\lim_{n\to\infty}\frac{n^2}{(n+1)^2}=|x|\left(\lim_{n\to\infty}\frac n{n+1}\right)^2=|x|<1$

so the sum will certainly converge whenever $-1<x<1$, but we also know the series will converge at the endpoints. $\displaystyle\sum\frac1{n^2}$ is a convergent $p$-series, which means $\displaystyle\sum\frac{(-1)^n}{n^2}$ is also (absolutely) convergent, so in fact the interval of convergence is $-1\le x\le1$ (endpoints included).