#### 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∞14+e−n
.

Here,
an=14+e−n
.

**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).

That is, compute the value of
limn→∞an=limn→∞14+e−n
.

limn→∞14+e−n=limn→∞1limn→∞4+limn→∞e−n=14+e−∞=14+0=14

Thus, the limit of the sequence is
limn→∞11+e−n=14
.

Since
limn→∞11+e−n≠0
and by using the Theorem (Series test for Divergence), the series
∑n=1∞11+e−n
is divergent.

Therefore, the series is divergent.