How do you test the alternating series $Sigma (-1)^n/lnn$ from n is $[2,oo)$ for convergence?

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Answer 1 · 2017-11-04T11:01:30

By the alternating series test criteria, the series converges

Suppose that we have a series $suma_n$ and either

$a_n=(-1)^nb_n$ or $a_n=(-1)^(n+1)b_n$ where $b_n>=0$ for all n.

Then if,

$1$ $lim_(n->oo)b_n=0$

and,

$b_n$ is a decreasing sequence

the series $suma_n$ is convergent.

Here, we have

$sum_(n=2)^oo(-1)^n/lnn=sum_(n=2)^oo(-1)^n/lnn=sum_(n=2)^oo(-1)^n*1/lnn$

$b_n=1/lnn$

$lim_(n->oo)b_n=lim_(n->oo)(1/lnn)=0$

So, by the alternating series test criteria, the series converges

How do you test the alternating series Sigma (-1)^n/lnnSigma (-1)^n/lnn from n is [2,oo)[2,oo) for convergence?