Bessel function values

Question

Given

$$J_m(x)=\sum_{n=0}^{\infty}{{(-1)^n}\over{n!(n+m)!}}\left(\frac{x}{2}\right)^{m+2n},$$ where $m=0,1,2,\ldots$ and $x\ge0$.

Need to show $$\left|J_m(x)\right|\le1.$$

Answer 1

I don't know if you're allowed to, but using that $$J_m(x)=\frac 1 \pi\int_0^\pi \cos(mt-x\sin t)dt$$

does the job.