Superiority of Root Test over Ratio Test

Suppose that is a sequence of strictly positive terms
with *a*(0) = 1. Put .The weak form of the ratio test says that if

Using elementary properties of logarithms these tests can be reformulated to say:

**Ratio Test:**
If

**Root Test:** If

Since

we have so if then
So the weak form of the root test will decide the convergence/divergence
of *A*(*n*) whenever the weak form of the ratio test does, and potentially
will decide the convergence/divergence of *A*(*n*) when the weak form of the
ratio test fails to give any information. The demonstration above makes it
clear why: The ratio test only uses one ratio, while the root test uses
all the ratios. One might say, the root test is better because it remembers it
roots.