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What is the radius of convergence of $$\sum_{n=1}^\infty \frac{nx^n}{n^2 + x^{2n}}?$$ Here is what I have so far. With the ratio test I want to find values of $x$ for which $$ \lim_{n\to\infty}x \frac{n+1}{n} \frac{n^2 + x^{2n}}{(n+1)^2 + x^{2(n+1)}} < 1.$$ If $x < 1$, this looks to be true. If $x = 1$, the series is essentially $\sum_{n=1}^\infty \frac{1}{n}$ so diverges. If $x = -1$, $\sum_{n=1}^n \frac{(-1)^n}{n}$ converges by the alternating series test. And for $x > 1$, the series seems to be essentially $$\sum_{n=1}^\infty \frac{1}{x^n}$$ which converges by the ratio test.

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    $\begingroup$ How do you define the radius of convergence for something that is not a power series? $\endgroup$ Jul 16 '17 at 4:13
  • $\begingroup$ Use root test I think. $\endgroup$
    – Crazy
    Jul 16 '17 at 4:16
  • $\begingroup$ I believe the root test is inconclusive. $\endgroup$ Jul 16 '17 at 4:21
  • $\begingroup$ use the ratio test $\endgroup$
    – BBot
    Jul 16 '17 at 4:27
  • $\begingroup$ Somewhat related: How to resolve this absolute value inequality $|1+x^2|>|x|$? $\endgroup$ Jul 16 '17 at 8:09
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Let's apply the ratio test

$$ \frac{a_{n+1}}{a_{n}} = x\frac{n+1}{n}\frac{n^2 + x^{2n}}{(n+1)^2+x^{2n+2}} =\frac{1}{x}\frac{n+1}{n}\frac{1 + (n/x^n)^2}{1+([n+1]/x^{n+1})^2} $$ If $|x| < 1$, then clearly the first expression goes to $x$ as $n\rightarrow \infty$, and if $|x|>1$, clearly the second expression goes to $x^{-1}$ as $n\rightarrow\infty$. So if $|x| \ne 1$, the series converges absolutely.

What if $x = \pm 1$? Then the series is $\sum_{n=0}^\infty (\pm 1)^n n/(n^2+1)$. If $x = 1$, this series diverges by the integral test against $f(t) = t/(t^2+1)$. If $x = -1$, this series converges by the alternating series test, though only conditionally.

So the series converges for all $x$ except $x = 1$, though only conditionally for $x = -1$.

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