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let-gt-0-prove-that-n-0-1-n-n-0-1-x-1-1-x-dx-

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Question Number 32026 by abdo imad last updated on 18/Mar/18
let α>0 prove that Σ_(n=0) ^∞ (((−1)^n )/(n+α)) =∫_0 ^1 (x^(α−1) /(1+x))dx .
letα>0provethat∑n=0∞(−1)nn+α=∫01xα−11+xdx.
Commented by abdo imad last updated on 22/Mar/18
∫_0 ^1 (x^(α−1) /(1+x))dx = ∫_0 ^1 (Σ_(n=0) ^∞ (−1)^n x^n )x^(α−1) dx = Σ_(n=0) ^∞ (−1)^n ∫_0 ^1 x^(n+α−1) dx =Σ_(n=0) ^∞ (−1)^n [ (1/(n+α)) x^(n+α) ]_0 ^1 =Σ_(n=0) ^∞ (((−1)^n )/(n+α)) .
∫01xα−11+xdx=∫01(∑n=0∞(−1)nxn)xα−1dx=∑n=0∞(−1)n∫01xn+α−1dx=∑n=0∞(−1)n[1n+αxn+α]01=∑n=0∞(−1)nn+α.

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