This is
WRONG. The root test is
SUPERIOR to the ratio test.
The actual full statement is based on
limsup and liminf. The test says given
. Consider
, the beauty here is that this limit exists (or
) not matter what, the reciprocal will be the radius of convergence. The ratio test is not superior, for one thing we require that the terms
be non-zero to use this test! While in root test this is not a necessity! Furthermore, the ratio test is based on root test, because of the fact that:
. Thus, if
then
and we have convergence. If
then
and we have divergence. All because of the root test. And finally we can give examples where there ratio test fails and root test succedes.