Consider the series where
for all
. Raabe’s test for convergence states that if there exists an integer
and a constant
such that
then the series is convergent. The series is divergent if
Proof. Assume there is an and an
such that
From the previous exercise (Section 10.16, Exercise #15) we know that if there exists an integer and an
such that
then converges. Taking
, we have
Hence, converges.
Next, if
then we have for all ,
Letting we then have
Since we know diverges, we may apply the previous exercise to conclude that
diverges