Infinity i 0 i2 4iSolutionStart with the geometric series 1

Solution

Start with the geometric series
1/(1 - x) = (i = 0 to ) x^i.

Differentiate both sides:
1/(1 - x)^2 = (i = 0 to ) ix^(i-1)

Multiply both sides by x:
x/(1 - x)^2 = (i = 0 to ) ix^i; note we have an i in from of x^n.

Repeat the process.
Differentiate both sides:
[(1 - x)^2 - x * -2(1 - x)]/(1 - x)^4 = (i = 0 to ) i^2 x^(i-1)
==> (1 + x)/(1 - x)^3 = (i = 0 to ) i^2 x^(i-1).

Multiply both sides by x:
x(1 + x)/(1 - x)^3 = (i = 0 to ) i^2 x^i.

Finally, let x = 1/4:
(1/4)(5/4) / (3/4)^3 = (i = 0 to ) i^2 (1/4)^i.
==> (i = 0 to ) i^2/4^i = 20/27.