Does $\sum_{n=1}^\infty \int_0^1 \frac{dt}{(n + t^2 x^2)^{3/2}}$ converge? For what values of $x$ does
$$\sum_{n=1}^\infty \int_0^1 \frac{dt}{(n + t^2 x^2)^{3/2}}$$
converge? Is the convergence absolute, conditional, or uniform?
I'm not sure how to begin solving this problem. Evaluating the integral doesn't
seem tractable. Interchanging summation and integration doesn't seem like it gives
anything. So I'm stuck.
 A: $$(n+t^{2}x^{2})^{\frac{3}{2}}\ge n^{\frac{3}{2}}$$
$$\implies\frac{1}{(n+t^{2}x^{2})^{\frac{3}{2}}}\le\frac{1}{n^{\frac{3}{2}}}$$
$$\implies\int_{0}^{1}\frac{1}{(n+t^{2}x^{2})^{\frac{3}{2}}}dt\le\frac{1}{n^{\frac{3}{2}}}$$
$$\implies\sum_{n=1}^{\infty}\int_{0}^{1}\frac{1}{(n+t^{2}x^{2})^{\frac{3}{2}}}dt\le\sum_{n=1}^{\infty}\frac{1}{n^{\frac{3}{2}}}$$
A: $\newcommand{\+}{^{\dagger}}
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\begin{align}
&\sum_{n = 1}^{\infty}\int_{0}^{1}{\dd t \over \pars{n + t^{2}x^{2}}^{3/2}}
=\sum_{n = 1}^{\infty}{1 \over n\root{n + x^{2}}}\
\color{#c00000}{\Large<}\
\sum_{n = 1}^{\infty}{1 \over n\root{n + 0^{2}}} =
\sum_{n = 1}^{\infty}{1 \over n^{3/2}}
\end{align}
