Although a source tells me that the series $\sum\limits_{n = 1}^{\infty} \dfrac{x}{n^p + x^2 n^q}$ is not uniformly convergent for $p > 1$ and $q \leq 0$ on any interval $\left[ a, b \right]$, I have tried the following:-
We have
$$\left| \dfrac{x}{n^p + x^2 n^q} \right| = \left| \dfrac{n^{-q} x}{n^{p - q} + x^2} \right| \leq \dfrac{n^{-q} \left| x \right|}{n^{p - q}} \leq \dfrac{\max \left\lbrace \left| a \right|, \left| b \right| \right\rbrace}{n^p}$$
Now, the series $\sum\limits_{n = 1}^{\infty} \dfrac{\max \left\lbrace \left| a\ \right|, \left| b \right| \right\rbrace}{n^p}$ is convergent for $p > 1$ so that the original series is uniformly convergent by Wiestrauss M test.
Is this proof correct and the answer from the source wrong? Or is there a mistake in the proof?