# How do you determine the convergence or divergence of Sigma (-1)^(n+1)cschn from [1,oo)?

Nov 7, 2017

The series converges

#### Explanation:

Perform the ratio test

${a}_{n} = {\left(- 1\right)}^{n + 1} \csch n$

Therefore,

$| {a}_{n + 1} / {a}_{n} | = | \frac{{\left(- 1\right)}^{n + 2} \csch \left(n + 1\right)}{{\left(- 1\right)}^{n + 1} \csch n} |$

$= | \frac{\csch \left(n + 1\right)}{\csch} n |$

Then,

${\lim}_{n \to \infty} | {a}_{n + 1} / {a}_{n} | = {\lim}_{n \to \infty} | \frac{\csch \left(n + 1\right)}{\csch} n |$
c
$= {\lim}_{n \to \infty} | \frac{{e}^{n} - \frac{1}{e} ^ \left(n\right)}{{e}^{n + 1} - \frac{1}{e} ^ \left(n + 1\right)} |$

=lim_(n->oo)|(((e^n-1/e^n))/(e(e^n-1/e^n))|

$= \frac{1}{e}$

As,

${\lim}_{n \to \infty} | {a}_{n + 1} / {a}_{n} | = \frac{1}{e} < 1$

We conclude that the series converges ( absolutely) by the ratio test