Question

How do you determine if the series the converges conditionally, absolutely or diverges given `Sigma ((-1)^(n))/(lnn)` from `[1,oo)`?

Answer 1

The series is convergent.

Explanation:

We have that `a_n = ((-1)^n)/(log(n))` It is an alternating series.

This series is converget if

1) `lim_(n->oo) a_n = 0`
2) `abs(a_(n+1)) < abs(a_n)`

This series accomplishes both requirements so it is convergent.

Attached a plot of `S(n)=sum_(k=2)^n((-1)^k)/(logk)`