Sum 1 + 3/4 + 7/16 + 15/64 + 31/256 +.... to infinity

HI Suhas,

The general term of this series is:

Tr = (2^r-1)/4^r-1, where r = 1,2,3,4,.......,upto infinity.

So Tr = 4/2^r - 1/4^r-1.

So, these are two GP Series.

Summation from 1 to infinity of these series is:

S1 = 4/2*[1/(1 - 1/2)] = 4, and

S2 = 1/(1 - 1/4) = 4/3.

So the required sum,

S = S1 - S2 = 4 - 4/3 = 8/3.

Hope it helps.

Wish you all the best.

Regards,

Ashwin (IIT MadraS).