How do you determine the convergence or divergence of #sum_(n=1)^(oo) (-1)^(n+1)/n#?
The series is convergent and its sum is:
Leibniz's theorem states that a sufficient condition for the series with alternating signs:
to be convergent is that:
We can actually calculate its sum starting from the MacLaurin development of the function
and we can easily conclude that:
so that the MacLaurin series is: