Determine whether the series is convergent or divergent. If it is convergent, find its sum.
Whether the series is convergent or divergent and obtain the sum if the series is convergent.
The series is divergent.
The series is ∑n=1∞enn2.
Theorem used: Series test for Divergence
If limn→∞an does not exist or limn→∞an≠0, then the series ∑n=1∞an is divergent.
Obtain the limit of the sequence (the value of the term an as n tends to infinity).
Since ∞∞ is in an indeterminate form, apply L’Hospital’s rule,
Since the limit of the sequence limn→∞enn2 does not exist.
Therefore, the sequence is divergent.
Since limn→∞enn2≠0 and by using the Theorem (Series test for Divergence), the series ∑n=1∞enn2 is divergent.
Therefore, the series is divergent.