you know we have all these tests for convergence of a series, but it always made me wonder if there exists any other method of finding the sum of a series, like we have the geometric series in which we are able to find the sum by a simple formula, but are we able to find such a formula for any series?, would such a formula be unique to each particular series?, or to a group of such series?

for example [tex]\sum_{n=1}^{\infty}\frac {(-2)^{n}}{n+1}[/tex]

this isnt a geometric series then how would you find the sum of such series?

is it possible to derive a formula for any series that exist, or are there limitations or conditions that need to be satisfied?