the ratio and root tests

Faced with a series sum z_n, you want a quick verdict: does it converge or not? The ratio and root tests are two such verdicts, and both work by comparing the series to a geometric series — the simplest series we fully understand, sum r^n, which converges exactly when r < 1. The trick is to measure how fast the terms shrink. If, deep into the series, each term is roughly r times the previous one with r < 1, the series behaves like a convergent geometric series and converges absolutely.

The ratio test (d'Alembert) looks at the ratio of consecutive lengths. Compute L = lim |z_(n+1)| / |z_n|. If L < 1 the series converges absolutely; if L > 1 it diverges; if L = 1 the test is silent and tells you nothing. The root test (Cauchy) instead takes the n-th root of the n-th length: compute L = lim |z_n|^(1/n), or more carefully its limit superior. Again L < 1 gives absolute convergence, L > 1 gives divergence, L = 1 is inconclusive. As a worked instance, for sum z^n / n! the ratio is |z^(n+1)/(n+1)!| / |z^n/n!| = |z| / (n+1), which tends to 0 < 1 for every z, so e^z's series converges everywhere.

These tests are the engine that powers the radius of convergence. Apply the root test to a power series sum a_n (z - z_0)^n: the n-th term has length |a_n| |z - z_0|^n, whose n-th root is |a_n|^(1/n) |z - z_0|. This is below 1 precisely when |z - z_0| < 1 / limsup |a_n|^(1/n) — which is exactly the Cauchy-Hadamard formula for R. So the root test does not merely test a fixed series; applied with z as a variable, it carves out the entire disk where the series lives. The root test is the more powerful of the two (it always works when the ratio test does, and sometimes when it does not), but the ratio test is often easier to compute.