a problem on test of converges of two series 
Test the following series for convergence:
(a)$$ \sum_{n=1}^\infty \frac{(n+1)^n}{n^{n+5/4}}$$
(b)$$\sum_{n=1}^\infty \frac{\tan \left(\frac{1}{n} \right)}{\sqrt{n}}.$$

Which formula/test would be useful to solve this problem?
 A: Useful formulas/results here:


*

*$\lim\limits_{n\to +\infty}\left(1+\frac 1n\right)^n=e$;

*$\tan\left(\frac  1n\right)\overset{n\to +\infty}{\sim}\frac 1n$;

*$\sum\limits_{n=1}^{+\infty}\frac 1{n^{\alpha}}$ is convergent if and only if $\alpha>1$;

*If $0<u_n, v_n$, $u_n=(1+\varepsilon_n)v_n$, where $\lim\limits_{n\to+\infty}\varepsilon_n=0$, then $\sum\limits_{n=1}^{+\infty}u_n$ is convergent if and only if so is $\sum\limits_{n=1}^{+\infty}v_n$.

A: Hint: Raabe's test, a higher order ratio test, works nicely for both of these series. 
A: Use the Kummer's Test: 

Given a series $\sum_{i=1}^{\infty}u_i$ of positive terms $u_i$ and a sequence of finite positive constants $a_i$, let
  $$
\rho=\lim_{n\to\infty}\left(a_n\cdot\frac{ u_n}{u_{n+1}}-a_{n+1}\right).
$$
  
  
*
  
*If $\rho>0$, the series converges.
  
*If $\rho<0$ and the series $\sum_{n=1}^{\infty}1/a_n$ diverges, the series diverges.
  
*If $\rho=0$, the series may converge or diverge.

The test is a general case of Bertrand's test, the root test, Gauss's test, and Raabe's test. With $a_n=n$ and $a_{(n+1)}=n+1$, the test becomes Raabe's test. 
