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
.