What's the sum of this power series?

Question

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! :\

Answer 1

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 $n=k-1$.