Here's a similar derivation (assuming |x| < 1), applying the geometric summation twice in the last two steps, but presented a bit more visually: x + 2x^2 +

Here's a similar derivation (assuming |x| < 1), applying the geometric summation twice in the last two steps, but presented a bit more visually: x + 2x^2 + 3x^3 + 4x^2.. = x + x^2 + x^3 + x^4 + ... + x^2 + x^3 + x^4 + ... + x^3 + x^4 + ... + x^4 + ... = sum i from 0 to infinity (sum j=i to infinity (x^j)) = sum i from 0 to infinity ((x^i)/(1-x)) = 1/(1-x)^2