Verified answer

Start with the geometric series 1/(1 - t) = Σ(n = 0 to ∞) t^n, convergent for |t| < 1.

Differentiate both sides:

1/(1 - t)² = Σ(n = 1 to ∞) nt^(n-1), convergent for |t| < 1.

Let t = -x:

1/(1 + x)² = Σ(n = 1 to ∞) n(-1)^(n-1) x^(n-1), convergent for |-x| = |x| < 1.

(So, the radius of convergence is 1.)

I hope this helps!