#### 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
.