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 .