An annoying infinite series question

The question is as follows:

What is the limit of the infinite series 1- 2q + 3q^2 - 4q^3 + 5q^4 - ...?

Student X

If you examined things carefully, you will realize that the series 1 - 2q + 3q^2 - 4q^3 + 5q^4 - .…. is actually the derivative of the geometric progression q - q^2 + q^3 - q^4 + q^5 -.......... (wrt q)

The sum to infinity of the GP q - q^2 + q^3 - q^4 +q^5 -.......... is simply given by q/(1-(-q)) = q/ (1+q)

Which therefore means the sum to infinity of 1- 2q + 3q^2 - 4q^3 + 5q^4 - ... is equal to d/dq [q/(1+q)]= 1/(1+q)^2 (shown)

Note that this entire thing works if and only if |q|<1,  ie  -1<q<1 (criteria for convergence)

Hope this helps. Peace.

Best Regards,

Mr Koh