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Question Number 166436 by mnjuly1970 last updated on 20/Feb/22

Show that : Σ_(n=1) ^∞ (((−1)^( n) H_( n) )/n^( 2) ) = −(5/8) ζ (3 ) ■ m.n −−−−−−−−−

Showthat:∑∞n=1(−1)nHnn2=−58ζ(3)◼m.n−−−−−−−−−

Answered by qaz last updated on 20/Feb/22

Σ_(n=1) ^∞ (((−1)^n H_n )/n^2 ) =Σ_(n=1) ^∞ (−1)^(n−1) H_n ∫_0 ^1 x^(n−1) lnxdx =∫_0 ^1 ((ln(1+x)lnx)/(x(1+x)))dx =∫_0 ^1 ((ln(1+x)lnx)/x)dx−∫_0 ^1 ((lm(1+x)lnx)/(1+x))dx =−(1/2)∫_0 ^1 ((ln^2 x)/(1+x))dx+(1/2)∫_0 ^1 ((ln^2 (1+x))/x)dx =(2^(1−3) −1)ζ(3)+(1/8)ζ(3) =−(5/8)ζ(3)

∑∞n=1(−1)nHnn2=∑∞n=1(−1)n−1Hn∫01xn−1lnxdx=∫01ln(1+x)lnxx(1+x)dx=∫01ln(1+x)lnxxdx−∫01lm(1+x)lnx1+xdx=−12∫01ln2x1+xdx+12∫01ln2(1+x)xdx=(21−3−1)ζ(3)+18ζ(3)=−58ζ(3)

Answered by mnjuly1970 last updated on 20/Feb/22

Σ_(n=1) ^∞ (((−1)^( n+1) )/n) ∫_0 ^( 1) x^( n−1) ln(1−x)dx = ∫_0 ^( 1) {ln(1−x).(1/x)Σ_(n=1) ^∞ (((−1)^( n+1) x^( n) )/n)dx} = ∫_0 ^( 1) ((ln(1−x).ln(1+x ))/x) dx = ((−5)/8) ζ (3)

∑∞n=1(−1)n+1n∫01xn−1ln(1−x)dx=∫01{ln(1−x).1x∑∞n=1(−1)n+1xnndx}=∫01ln(1−x).ln(1+x)xdx=−58ζ(3)

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