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 .