- #1
DeadOriginal
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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!