ratio test
The ratio test asks how each term compares in size to the one before it. If, in the long run, every term is a fixed fraction smaller than its predecessor, the series behaves like a shrinking geometric series and converges; if each term tends to be larger, it blows up. It is the test of choice when terms involve factorials or exponentials.
Formally, consider L = lim |a_{n+1} / a_n| (using the limit superior if the ordinary limit fails to exist). Then: if L < 1 the series sum a_n converges absolutely; if L > 1 (or the ratios tend to infinity) the series diverges; if L = 1 the test is inconclusive and gives no information. The idea of the proof is exactly the geometric comparison: for L < 1 the tail is dominated by a convergent geometric series with ratio just above L.
Be candid about the L = 1 blind spot: every p-series has ratio limit 1, yet some converge (p > 1) and some diverge (p <= 1), so the ratio test cannot distinguish them. When L = 1, switch to the root test (which is at least as powerful), the integral test, or comparison. The ratio test shines precisely when the ratio has a clean limit away from 1.
For sum 1/n! the ratio |a_{n+1}/a_n| = n!/(n+1)! = 1/(n+1) -> 0 < 1, so the series converges (its sum is e - 1 from n = 1, or e from n = 0).
Factorials make the ratio test natural; here the ratio tends to 0.