What is the value of $1+\frac{1}{2^2}+ \frac{1}{4^2}+ \frac{1}{8^2}+... $? I need to find the value of the following series:
$$ 1+\frac{1}{2^2}+ \frac{1}{4^2}+ \frac{1}{8^2}+\cdots $$ 
That is 
$$
1+ \frac{1}{2^2}+ \frac{1}{(2^2)^2} +\frac {1}{(2^3)^2}+ \cdots$$
It is the summation of a geometric progression where all the terms are squared. I am unable to go further with it.  

Why is it that the squares give you trouble? Your series is simply$$1+\frac1{2^2}+\frac1{(2^2)^2}+\frac1{(2^2)^3}+\cdots$$and therefore its sum is$$\frac1{1-\frac1{2^2}}=\frac43.$$
