#### To determine

Whether the series is convergent or divergent and obtain the sum if the series is convergent.

#### Answer

The series is divergent.

#### Explanation

**Given:**

The series is ∑n=1∞enn2.

Here, an=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.

**Calculation:**

Obtain the limit of the sequence (the value of the term an as *n* tends to infinity).

limn→∞an=limn→∞enn2

Since ∞∞ is in an indeterminate form, apply L’Hospital’s rule,

limn→∞enn2=limn→∞ddn(en)ddn(n2)=limn→∞en2n

Since ∞∞ is in an indeterminate form, apply L’Hospital’s rule,

limn→∞enn2=limn→∞ddn(en)ddn(2n)=limn→∞en2=e∞2=∞

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.