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 .