# Convergence of $\sum_{n=1}^{\infty}\frac{(n+1)^n}{n^{n+\frac3{2}}}$

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.

• Expontential is hiding ... – Isham Sep 22 '17 at 13:10

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 $a>1$.
• 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? – vidyarthi Sep 22 '17 at 13:09
• Note that $(1+\frac1{n})^n$ is an increasing sequence which tends to $e$. – Robert Z Sep 22 '17 at 13:11