Chapter 10: Infinite Sequences and Series - Section 10.10 - The Binomial Series and Applications of Taylor Series - Exercises 10.10 - Page 633: 51

$\dfrac{-1}{(1+x)^2} ; |x| \lt 1 $

Recall the Taylor series for $\dfrac{1}{1-x}= 1+x+x^2+......+x^n+.....; |x| \lt 1$ Now, $=-1+2x-3x^2+4x^3-.....$ or, $=\dfrac{d}{dx} [1-x+x^2-x^3+x^4-......]$ or, $=\dfrac{d}{dx} (1/1+x) $ or, $=\dfrac{-1}{(1+x)^2} ; |x| \lt 1 $