Calculate $\sum_{n=1}^\infty\int_{\sqrt{3+n^{1/5}}}^{\sqrt{4+n^{1/5}}}\frac{e^{t^2}}{(e^{t^2}+1)t^2}dt$.

Question

Check the convergence of the series $$\sum_{n=1}^\infty\int_{\sqrt{3+n^{1/5}}}^{\sqrt{4+n^{1/5}}}\frac{e^{t^2}}{(e^{t^2}+1)t^2}dt$$ Any suggestions please? I can't calculate integral!

Thanks in advance.

Answer 1

We have: $$\sqrt{4+n^{1/5}}-\sqrt{3+n^{1/5}}=\frac{1}{\sqrt{4+n^{1/5}}+\sqrt{3+n^{1/5}}}\gg\frac{1}{n^{1/10}}$$ and the integrand function $f(t)$ is positive and decreasing over $\mathbb{R}^+$. Since: $$f\left(\sqrt{4+n^{1/5}}\right)\geq\frac{1}{2(4+n^{1/5})}\gg\frac{1}{n^{1/5}}$$ it follows that: $$\sum_{n=1}^{N}\int_{\sqrt{3+n^{1/5}}}^{\sqrt{4+n^{1/5}}}f(t)\,dt \gg \sum_{n=1}^{N}\frac{1}{n^{3/10}},$$ but the RHS is a divergent series.