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
253
.
Explanation
Given:
The series is
10−2+0.4−0.08+⋯
.
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=10
.
The common ratio of the series is,
r=Second termFirst term=−210
The absolute value of r is,
|r|=|−210|=210=0.2<1
Since
|r|<1
and by using the Result stated above, the series is convergent.
Obtain the sum of the series.
Since
a=10 and r=−210
,
10−2+0.4−0.08+⋯=101−(−210)=101+210=1010+210
Perform the arithmetic operations and simplify the terms,
10−2+0.4−0.08+⋯=1011210=101×1012=5×512=253
Therefore, the series is convergent to the sum
253
.