Next: About this document ...

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

then

while if

then

The weak form of the root test says that if

then

while if

then

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

Ratio Test: If

then

while if

then

Root Test: If

then

while if

then

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.

Next: About this document ...
Eric S Key
3/23/1999