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! :\
What's the sum of this power series?
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special-functions
power-series
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1This is related to a Bessel function; see http://en.wikipedia.org/wiki/Bessel_function#Definitions – 2011-04-06
1 Answers
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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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0@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