I saw a post asking for the solution of $\int x^n e^x dx$ for an arbitrary positive integer $n.$
(How to integrate $ \int x^n e^x dx$?) In light of that, how does one determine a general solution in terms of a series?
Thank you!!!!
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3Do integration by parts – Amer Oct 31 '19 at 17:16
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1Why do you want a series? It's an elementary integral, which doesn't necessarily need a series to write. – bjorn93 Oct 31 '19 at 17:31
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After giving n different values and doing integration by parts, I was simply interested in how the integral could be represented by a series. – bonobobo Oct 31 '19 at 17:38
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\begin{align} & \int x^ne^x\,\mathrm dx =\int x^n\sum_{k\ge0}\frac{x^k}{k!}\,\mathrm dx \\[8pt] ={} & \int\sum_{k\ge0}\frac{x^{n+k}}{k!}\,\mathrm dx = \sum_{k\ge0}\frac1{k!}\int x^{n+k}\,\mathrm dx \\[8pt] = {} & C+\sum_{k\ge0}\frac{x^{n+k+1}}{k!(n+k+1)} \end{align}
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