Mathematics

Problem

The inverse-tangent function satisfies

\arctan x =\sum_{n=0}^{\infty} (-1)^n \dfrac{x^{2n+1}}{2n+1}

for

|x|\le 1

. Which of the following gives a power-series representation for

\dfrac{1}{1+x^{2}}

valid for

|x|<1

? a)

\displaystyle \sum_{n=0}^{\infty} (-1)^n x^{n}

\displaystyle \sum_{n=0}^{\infty} (-1)^n (2n+1) x^{2n}

\displaystyle \sum_{n=0}^{\infty} (-1)^n \dfrac{x^{2n}}{2n+1}

\displaystyle \sum_{n=0}^{\infty} (-1)^n x^{2n}