First consider when
.
For
, we have
, which implies
and
.
By the comparison test, we have
Therefore, for
, the given series diverges.
On the other hand, for
, note that
is continuous, positive, and decreasing on
.
In this case, we'll use the integral test.
For
but
, we have
provided that
,
Here, the second equality is obtained from the substitution
.
Using
,
Then, by the integral test, the given series is convergent for
; ,
For
, the integral can be computed by
.
Therefore, again by the integral test, the given series diverges when
.
To sum, the answer is L.