Identify whether the series summation of 12 open parentheses 3 over 5 close parentheses to the i minus 1 power from 1 to infinity is a convergent or divergent geometric series and find the sum, if possible. identify whether the series summation of 12 open parentheses 3 over 5 close parentheses to the i minus 1 power from 1 to infinity is a convergent or divergent geometric series and find the sum, if possible.

Given series is sigma i=1 to infinity 12*,

                                       =

Clearly it is a geometric series and it converges if and only if r<1

So, common ratio = < 1

Hence the series is convergent.

Formula for sum of infinite geometric series =

Where a is the first term and r is the common ratio.

So, sum of series =

For taking 2/5 to numerator we have to multiply with reciprocal 5/2 on both numerator and denominator.

Hence sum of series = 12* = 30