Take two sequences and with for all beyond some point . Now if the series diverges then the series does too, and if the series converges to then the series of converges to .
[UPDATE]: I overstated things a bit here. If the series of converge, then so does that of , but the inequality only holds for the tail beyond . That is:
but the terms of the sequence before may, of course, be so large as to swamp the series of .
If we have two nonnegative sequences and so that then the series and either both converge or both diverge.
We read in Cauchy’s condition as follows: the series converges if and only if for every there is an so that for all the sum .
We also can import the notion of absolute convergence. We say that a series is absolutely convergent if the series is convergent (which implies that the original series converges). We say that a series is conditionally convergent if it converges, but the series of its absolute values diverges.