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
83
.
Explanation
Given:
The series is
2+0.5+0.125+0.03125+⋯
.
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 first term of the series is
a=2
.
The common ratio of the series is,
r=Second termFirst term=0.52=0.25
The absolute value of r is,
|r|=|0.25|=0.25<1
Since
|r|<1
and by using the Result stated above, the series is convergent.
Obtain the sum of the series.
Since a = 2 and r = 0.25,
2+0.5+0.125+0.03125+⋯=21−0.25=20.75×100100=20075=83
Therefore, the series is convergent to the sum
83
.