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Question Number 144928 by ArielVyny last updated on 30/Jun/21

Σ_(n≥0) (((−1)^n )/((n+z)n!))

∑n⩾0(−1)n(n+z)n!

Answered by Dwaipayan Shikari last updated on 01/Jul/21

∫_0 ^1 Σ_(n≥0) (((−1)^n )/(n!))x^(n+z−1) dx =∫_0 ^1 e^(−x) x^(z−1) dx=[−e^(−x) x^(z−1) ]_0 ^1 +(z−1)∫_0 ^1 e^(−x) x^(z−2) dx I(z−1)=−(1/e)+(z−1)I(z−2) ... Or ∫_0 ^1 e^(−x) x^(z−1) dz=Γ(1,z) (Incomplete Gamma function)

∫01∑n⩾0(−1)nn!xn+z−1dx=∫01e−xxz−1dx=[−e−xxz−1]01+(z−1)∫01e−xxz−2dxI(z−1)=−1e+(z−1)I(z−2)...Or∫01e−xxz−1dz=Γ(1,z)(IncompleteGammafunction)

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