#### 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∞11+(23)n
.

Here,
an=11+(23)n
.

**Result used:**

The sequence
{sn}
is converges to zero when
−1<s<1
.

That is,
limn→∞sn=0 if −1<s<1
.

**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→∞11+(23)n
.

limn→∞11+(23)n=limn→∞1limn→∞1+limn→∞(23)n=11+limn→∞(23)n

Since
−1<23<1
and by using the Result stated above, the value of
limn→∞(23)n=0
.

limn→∞11+(23)n=11+0=11=1

Thus, the limit of the sequence is
limn→∞11+(23)n=1
.

Since
limn→∞11+(23)n≠0
and by using the Theorem (Series test for Divergence), the series
∑n=1∞11+(23)n
is divergent.

Therefore, the series is divergent.