Limit comparison test
Suppose that we have two series and with for all .
Then if with then either both series converge or both series diverge.
Because we know that for all there is an integer such that for all we have that , or what is the same
As we can choose to be sufficiently small such that is positive. So and by the direct comparison test, if converges then so does .
Similarly , so if converges, again by the direct comparison test, so does .
That is both series converge or both series diverge.
We want to determine if the series converges. For this we compare with the convergent series .
As we have that the original series also converges.