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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?

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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 $\sum\limits_{n=1}^{+\infty}v_n$.
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+1 for indicating useful tools rather than writing down a full proof. – Did Jan 4 '13 at 15:06
@did See the Kummer test in – MathOverview Jan 4 '13 at 23:09
@Elias Listen, I am flattered by the attention, but why do you see fit to provide me with a reference to Kummer's test? – Did Jan 4 '13 at 23:23

Hint: Raabe's test, a higher order ratio test, works nicely for both of these series.

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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 $$ \rho=\lim_{n\to\infty}\left(a_n\cdot\frac{ u_n}{u_{n+1}}-a_{n+1}\right). $$

  1. If $\rho>0$, the series converges.

  2. If $\rho<0$ and the series $\sum_{n=1}^{\infty}1/a_n$ diverges, the series diverges.

  3. 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.

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