#### 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∞2+n1−2n
.

Here,
an=2+n1−2n
.

**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→∞2+n1−2n
.

Divide the numerator and the denominator by the highest power.

limn→∞2+n1−2n=limn→∞2+nn1−2nn=limn→∞2n+nn1n−2nn

Redefine the terms as given below.

limn→∞2+n1−2n=limn→∞2n+11n−2=limn→∞2n+limn→∞1limn→∞1n−limn→∞2=0+10−2=−12

Thus, the limit of the sequence is
limn→∞2+n1−2n=−12
.

Since
limn→∞2+n1−2n≠0
and by using the Theorem (Series test for Divergence), the series
∑n=1∞2+n1−2n
is divergent.

Therefore, the series is divergent.