Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.
Whether the geometric series is convergent or divergent and obtain the sum if the series is convergent.
The series is convergent to the sum
The series is
The geometric series
∑n=1∞arn−1 (or) a+ar+ar2+⋯
is convergent if
and its sum is
, where a is the first term and r is the common ratio of the series.
Consider the given series
Here, the first term of the series is
and the common ratio of the series is,
r=Second termFirst term=5π25π=5π2⋅π5=1π
The absolute value of r is,
and by using the Result stated above, the series is convergent.
Obtain the sum of the series.
Therefore, the series is convergent to the sum