## Convergence Tests for Infinite Series

Now that we’ve seen infinite series as improper integrals, we can immediately import our convergence tests and apply them in this special case.

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.

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