Determine whether the geometric series is convergent or divergent.
9 + 8 + 64/9 + 512/81 +…. If it is convergent, find its sum?

Solution:

Given: Geometric series is 9 + 8 + 64/9 + 512/81 +….

The given series looks like to be in geometric progression with constant common ratio r

r = 2^nd term/1^st term = a₂/a₁

= 8/9

We know that if the ‘r’ is less than 1 then it is convergent

Here 8/9 < 1.

So, the series is convergent.

Clearly, this is the sum of an infinite geometric series.

sum of GP = a/(1 - r)

Where a is the first term, 4 is the common ratio.

Sum = 9/(1-8/9)

= 9/(1/9)

Sum = 81

Therefore, the given series is convergent and its sum is 81.

Summary:

The geometric series 9 + 8 + 64/9 + 512/81 +…. is convergent and its sum is 81.