The Root Test
The root test gives information about the convergence or not of a series by considering
Theorem (The Root Test).
If there exists some N such that
for some q with n>=N thenconverges absolutely. If for infinitely many
thendiverges.
Proof: Suppose there is a real number
such that
and a positive number
such that for
then
and since
converges, hence by the comparison test
converges absolutely.
Suppose now that
for infinitely many
then for infinitely many
so 
does not converge to zero anddoes not converge.
Example:
even,
odd. 
Even and
odd. Thus
hence by the root test the series converges absolutely.
Example: Does
converge for
so
converges to 1 so
converges to
Let
then
so there is a positive integersuch that
implies that
thusconverges absolutely.
The root test has wider scope than the ratio test, since if the root test gives no results for convergence, then neither does the ratio test. Sometimes however the root test can be hard to use, and more information can usefully be obtained from the ratio test. Neither root nor ratio test will identify series that converge conditionally.