#### 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
.