TITLE: which $p,q$ makes the following function continuous
QUESTION [3 upvotes]: Let $p,q > 0$ and $\max \{ p,q \} > 1$. Let $ f(x) = \sum _ { n =1 } ^ \infty \frac { x } { n^p + x ^2 n ^q } $. The problem asks which $p,q $ makes $f$ continuous on all of $\mathbb R$.
I know if the series converges uniformly on $\mathbb R$ then $f$ is continuous. But I don't know how to associate this to the values of $p,q$. Any help is appreciated.

REPLY [0 votes]: Here is an approach I would take: Hopefully the case for $p=q$ should be clear, and for the other cases, I would try the ratio test.
Consider $a_n=\frac{x}{n^p+n^qx^2}$, and we know that the radius of convergence is infinity if $\lim_{n\rightarrow\infty}\lvert\frac{a_{n+1}}{a_n}\rvert=0$. Notice that
$$\frac{a_{n+1}}{a_n}=\frac{1}{\frac{(n+1)^p}{n^p}+\frac{(n+1)^q}{n^p}x^2}+\frac{x^2}{\frac{(n+1)^p}{n^q}+\frac{(n+1)^q}{n^q}x^2},$$
and compute the limit as $n\rightarrow\infty$ for the cases $p>q,p<q$.
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