To work with factorial in series, you often simply need the ratio test. However, sometimes the ratio test doesn’t give a tractable fraction. In those cases it is good to remember the definition of factorial, in particular the fact that n! = n·(n-1)!, and that tests for convergence typically have conditions that need only hold eventually.

1. For example, the sum from n=1 to infinity of (n!)/(n^n). Once n is at least 2, we can compare to a nice series we know about.

The sum from n=1 to infinity of 2/(n^2) is easily shown to converge, and, from a finite point on, it sits on top of our original series, which has only positive terms and hence must also converge.

2. We could use the ratio test for the sum , but we can also use the alternating series test. A similar trick to the previous example shows the magnitudes of the terms have limit 0. Once n is at least 3, we have the following comparison on the absolute value of each term.

That last fraction has a limit of 0 as n goes to infinity.

However, for the alternating series test we need more than a limit of zero; the magnitudes must decrease to 0, for which we use the fact that (n+1)! can be written in terms of n!:

As long as n is at least 3, the n+1st term is the nth term times a value less than 1. The series doesn’t decrease right from the start, but from a finite point on it always decreases, and that is enough.

3. How about the sum from n=1 to infinity of ? You might be tempted to use the ratio test for this example, but it would waste your time. The terms simplify to [(n+1)(n+2)]/n^2, and have a limit of 1 as n goes to infinity, and therefore the series diverges by the test for divergence.