#### To determine

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

#### Answer

The series is convergent to the sum
5π−1
.

#### Explanation

**Given:**

The series is
∑n=1∞5πn
.

**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:**

Consider the given series
∑n=1∞5πn
.

∑n=1∞5πn=5π1+5π2+5π3+⋯=5π+5π2+5π3+⋯

Here, the first term of the series is
a=5π
and the common ratio of the series is,

r=Second termFirst term=5π25π=5π2⋅π5=1π

The absolute value of *r* is,

|r|=|1π|=1π≈0.3183<1

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

Obtain the sum of the series.

Since
a=5π
and
r=1π
.

∑n=1∞5πn=5π1−1π=5ππ−1π=5π⋅ππ−1=5π−1

Therefore, the series is convergent to the sum
5π−1
.