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## Homework Statement

I want to show that

$$

\tan^{-1}(x)=\sum\limits_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}x^{2n+1}.

$$

## Homework Equations

I start with

$$

\int\frac{1}{1+x^{2}}dx.

$$

## The Attempt at a Solution

I want to be able to do the following:

$$

\int\frac{1}{1+x^{2}}dx=\int\sum\limits_{n=0}^{\infty}(-1)^{n}x^{2n}dx=\sum\limits_{n=0}^{\infty}\int (-1)^{n}x^{2n}dx

$$

but I am afraid that the infinite sum might create problems. Can anyone take a look? Thanks!