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How could we test the convergence of the following series?$$\sum_{n=1}^{\infty}\frac{(n+1)^n}{n^{n+\frac3{2}}}.$$

The root test clearly fails and hence the ratio test. With what should I compare the series to? Is Dirichlet's or, Abel's test helpful? Thanks beforehand.

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  • $\begingroup$ Expontential is hiding ... $\endgroup$ – Isham Sep 22 '17 at 13:10
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Hint. Note that $$\frac{(n+1)^n}{n^{n+\frac3{2}}}=\frac{(1+1/n)^n}{n^{3/2}}< \frac{e}{n^{3/2}}$$ Recall that $\sum_n 1/n^a$ is convergent iff $a>1$.

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  • $\begingroup$ I noticed the resemblance to $e$. But, the fact is that the sequence $(1+\frac1{n})^n$ is inside the series symbol, which confuses me. Could you be more rigorous? $\endgroup$ – vidyarthi Sep 22 '17 at 13:09
  • $\begingroup$ ok, understood. The series is bounded above, so trivial! $\endgroup$ – vidyarthi Sep 22 '17 at 13:11
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    $\begingroup$ Note that $(1+\frac1{n})^n$ is an increasing sequence which tends to $e$. $\endgroup$ – Robert Z Sep 22 '17 at 13:11
  • $\begingroup$ Apart from this, does dirichlet, kummer or abel tests work? $\endgroup$ – vidyarthi Sep 22 '17 at 13:12
  • $\begingroup$ @vidyarthi Abel test works $\endgroup$ – Robert Z Sep 22 '17 at 13:14

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