This question is similar to the one in: The error for approximation of bessel function

I know that for any integer $j{\geq}0$, $$\mathbb{I}_{j}(z) = \left(\frac{z}{2}\right)^j\sum_{k=0}^{\infty}\frac{z^{2k}}{4^{k}k!\Gamma(k+j+1)}.$$

I am trying to find a $K$ for which, given $x$, I can bound the truncation error to some $\epsilon$. That is, I want to find $K$, such that $$\left(\frac{z}{2}\right)^j\sum_{k=K+1}^{\infty}\frac{z^{2k}}{4^{k}k!\,\Gamma(k+j+1)} \leq \epsilon.$$

Is there an easy way to do this? Thanks!



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