#### To determine

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

#### Answer

The series is convergent to the sum 9.

#### Explanation

**Given:**

The series is
∑n=1∞3n+14−n
.

**Result used:**

The geometric series
∑n=1∞arn−1 (or) a+ar+ar2+⋯
is convergent if
|r|<1
and its sum is
a1−r
, where *a* is the first term and *r* is the common ratio of the series.

**Calculation:**

The given series can be written as follows,

∑n=1∞3n+14−n=∑n=1∞3n+14n=31+141+32+142+33+143+34+144+⋯=324+3342+3443+3544+⋯=94+2716+8164+243256+⋯

Clearly, it is geometric series.

The first term of the series is,
a=94
and the common ratio of the series is,

r=Second termFirst term=271694=2716⋅49=34

The absolute value of *r* is,

|r|=|34|=34=0.75<1

Since
|r|<1
and by the Result stated above, the series is convergent.

Obtain the sum of the series.

Since
a=94
and
r=34
,

∑n=1∞3n+14−n=941−34=9414=94⋅41=9

Thus, the sum of the series is
∑n=1∞3n+14−n=9
.

Therefore, the series is convergent to the sum 9.