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SumThePrimes

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I have recently come up with a fascinating method of summing various divergent series, whether it has been done before I am unaware, but in some scenarios it involves the anti-derivative of e^(-x)*sin(x)*x^n with respect to x. I only need positive integer values of n, and of course a proof/explanation. Any help would be appreciated. Forgive me if this is in the wrong place or is phrased awkwardly... I also accidently hit the enter key before I was done... It does involve number theory at it's heart, but that was not presented in the problem...

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