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prove equality with integral and series
This integral was my homework question with $p=2$ and $n=1$. I am wondering if one can get the general formula for p, or at least approximate this integral for any $p$, $n=1$? $ \int_{0}^{\infty}\left(\frac{2^n}{t^n}\left(\frac{t^n}{2^nn!}-\frac{1}{2^{n+2}}\frac{t^{n+2}}{1!(n+1)!}+\frac{1}{2^{n+4}}\frac{t^{n+4}}{2!(n+2)!}-\ldots\right)\right)^ptdt $
Thank you.