It seems that the following limit exists. But I couldn't figure out the exact value. Anyone could help me? Thanks!
\begin{align*}
\lim_{t\rightarrow 0^{+}} {\sum_{n=1} ^{\infty} \frac{\sqrt{t}}{1+tn^2}}
\end{align*}
Answer
Hint:
$$
\sqrt t \int_1^\infty {\frac{1}{{1 + tx^2 }}\,dx} = \int_{\sqrt t }^\infty {\frac{1}{{1 + x^2 }}\,dx} \to \frac{\pi }{2}.
$$
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