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Let $\sum_{n=0}^\infty c_n x^{2n}$ be the Taylor series of $f(x)=\frac{1}{1+x^2}$ at $x=0$ (the series contains only even powers of $x$ since $f(x)$ is an even function).

What is the expression for the $n$th coefficient $c_n$?

$1$

$(-1)^n$

$\cfrac{1}{n!}$

$\cfrac{(-1)^n}{n!}$

$\cfrac{2^n}{(2n)!}$