Solution to Infinite Sum?

Question

I can't seem to figure out a way to solve this:

Summation from i=0 to infinity of i^2 / 4^i. I computed the answer to be 20/27 using a program, but I'm looking for an analytical solution (steps), not just a number. Any suggestions as to how I can approach this are appreciated.

Answer 1

If you are familiar with term-by-term differentiation of series, it might be the quickest way to do it.

The geometric series asserts that the sum from i = 0 to infinity of x^i is 1/(1-x).  (At least for |x| < 1.)

The derivative of the right hand side with respect to x is -1/(1-x)^2.  By some theorems on the differentiation of series term-by-term the derivative of the left hand side is the sum from i = 0 to infinity of i x^{i-1}.  So

sum i=0..infinity i x^{i-1} = -1/(1-x).  (at least for |x| < 1)

If we multiply both sides by x, the x distributes across the infinite sum and we get

sum i=0..infinity i x^i = x/(1-x)^2.

Now differentiate again.  the left hand side has derivative

sum i=0...infinity i*i*x^{i-1}.  

The right hand side has derivative (after a lot of algebra)  (x+1)/(1-x)^3.  If we equate these and multiply by x, we get

sum i=0..infinity i^2 x^i = x(x+1)/(1-x)^3.

If you plug in x = 1/4, the left hand side is the series you want, and the right is indeed 20/27.

The power series method is pretty handy if you are allowed to use it.  If not, perhaps there is some algebraic method of arriving at the above general formulas for sum i x^i and i^2 x^i once you know what the right answer is.  Good luck.