can-some-one-find-the-exact-value-of-n-0-1-n-2-

Question Number 206890 by mathzup last updated on 29/Apr/24

c a n s o m e o n e f i n d t h e e x a c t v a l u e o f

\sum_{n = 0}^{\infty} \frac{1}{{(n!)}^{2}}

Commented by Frix last updated on 30/Apr/24

This is a Modified Bressel Function of the

First Kind .

I_{n} (z) = \frac{1}{π} \underset{0}{\int^{π}} e^{z \cos θ} \cos n θ d θ

With n = 0 this is equal to

I_{0} (z) = \underset{k = 0}{\sum^{\infty}} \frac{{(\frac{z^{2}}{4})}^{k}}{{(k!)}^{2}}

\Rightarrow \underset{k = 0}{\sum^{\infty}} \frac{1}{{(k!)}^{2}} = I_{0} (2) \approx 2.27958530234

I found this in a script, without proof, so

{you}^{'} ll have to do some research \dots

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