I don't know who downvoted this, but it's correct.

The use of series is a little "sloppy", but x + 2 x^2 + 3 x^3 + ... has absolute uniform convergence when |x|<r<1, even more importantly that it's true even for complex numbers |z|<r<1.

The super nice property of complex analysis is that you can be almost ridiculously "sloppy" inside that open circle and the Conway book will tell you everything is ok.

[I'll post a similar proof, but mine use -1/10 and rounding, so mine is probably worse.]