Does this series converge or diverge?

Question

summation of
[(-1)^(n-1)] / ln (n+1)
from 1 to infinity

please show work

Answer 1

It's an alternating series, for which the magnitude of the terms decreases with n (you should show this -- easy) and with the nth term going to zero as n -> infinity (also easy to show). Therefore, the series converges.

Answer 2

summation of the factors: convergent

Examining the numerator, it is either 1 or -1 for each value of n.
The denominator will slowly increase in value with n meaning the value of the ratio will decrease over time and have less impact on the summation of the individual values. In fact the series itself converges to 0 as n approaches infinity.

There is a theorem in calculus that states that the kth term of a convergent series tends to zero and that is what we have here.