If 0<x<1, find the following sum: S = 1+2*x + 3*x^2 + 4*x^3 + ...

The first thought when trying to solve such a problem is that you might be able to write this sum as a geometric progression. Luckily, it is the case here as well, as we can observe that S is the derivative (with respect to x) of another sum: P = x + x^2 + x^3 + ... . We can easily find P = x * (1-x^N)/(1-x), where N tends to infinity so P reduces to P = x/(1-x). Now, in order to calculate S, we ca simply take the first order derivative of P and find that S=1/(1-x)^2.