Alternate Series Test (Leibniz's Test)

Not sure why this one is considered so significant, but I found two texts that name it, including one that attaches the name of a true luminary, so here it is:

If {aⁿ} is a decreasing sequence that converges to 0, then Σ(-1)ⁿ⁺¹aⁿ converges.

Unfortunately, neither of those sources indicated why this was in any way important. It seems fairly obvious and the applications are limited. Digging a bit more, I found a generalization attributed to Dirichlet, another luminary (even bigger than Leibniz in the area of Bayesian Data Analysis). I rather like this one:

If {aⁿ} is a monotone decreasing real sequence converging to zero and {bⁿ} is a complex sequence such that



for all N where M is a constant, then Σaⁿbⁿ converges. This isn't just a little more general than Leibniz's Test. It allows you to use any sequence (real or complex) with a bounded sum to throttle the decreasing sequence and come up with a convergent sum.

And, Dirichlet either didn't know or didn't care that it was important. It wasn't published until after his death.