What is the sum of the reciprocals of this infinite series involving powers of 2 and squares?

Question

I can't seem to find this solution online. What is the sum of the reciprocals of the infinite series 2 to the power of squares? If I'm asking that right... I mean 1/(2^(n^2)) + 1/(2^((n+1)^2)) + ... The number series should go 1/1 + 1/2 + 1/16 + 1/512 + 1/65,536...

Or does it diverge and doesn't have a single sum?

Edit: I found a calculator to do it. The sum is .56446841360593857933...

Answer 1

This is a geometric series with ratio of a=1/2 and r=1/2.

The sum is a/(1-r), i.e. 1.

S = 1/2 + 1/4 + 1/8 ......

(1/2)= 1/4 + 1/8+....

Subtract

(1/2)S = 1/2 and as stated S = 1