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What's the sum of this power series? $f_k(x)=1-\frac{x^2}{k}+\frac{x^4}{k(k+1)\cdot2!}-\frac{x^6}{k(k+1)(k+2)\cdot3!}+\ldots$ I'm just helping someone, I'm not good at math! :\

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    This is related to a Bessel function; see http://en.wikipedia.org/wiki/Bessel_function#Definitions2011-04-06

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To expand on my comment: Your function is

$f_k(x)=(k-1)!\sum_{m=0}^\infty \frac{(-1)^m}{m!(k+m-1)!}x^{2m}\;.$

The Bessel function of (integer) order $n$ is

$J_n(x)=\sum_{m=0}^\infty\frac{(-1)^m}{m!(m+n)!}\left(\frac{x}{2}\right)^{2m+n}\;.$

Thus your function is

$f_k(x)=n!J_n(2x)x^{-n}\;,$

with $n=k-1$.

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    @Mitch: I think it depends on the question. In this case, there wasn't much variation in possible answers, and so Fabian's answer turned out to be not much different from a part of mine. I agree with you in cases where different answers can throw light on different aspects of the problem.2011-04-06