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Question Number 123687 by mnjuly1970 last updated on 27/Nov/20
...nice calculus.. prove that:: lim_(x→0) (((2φ(x))/x^2 ) +(π^2 /(3x))) =^(???) ζ(3) where φ(x)=∫_0 ^( 1) (((t^x −1)(ln(1−t)))/(tln(t)))dt
…nicecalculus..provethat::limx→0(2ϕ(x)x2+π23x)=???ζ(3)whereϕ(x)=∫01(tx−1)(ln(1−t))tln(t)dt
Answered by mnjuly1970 last updated on 28/Nov/20
solution: φ(x)=−∫_0 ^( 1) {((t^x −1)/(ln(t)))Σ_(n=1) ^∞ (t^(n−1) /n)}dx =−Σ_(n=1 ) ^∞ (1/n)∫_0 ^( 1) ((t^(x+n−1) −t^(n−1) )/(ln(t)))dt =−Σ_(n≥1) (1/n)[ln(x+n)−ln(n)] =−Σ_(n≥1) (1/n)ln(((x+n)/n))=−Σ_(n≥1) (1/n)ln(1+(x/n)) =−Σ_(n≥1) (1/n)Σ_(m≥1) (((((−x)/n))^m )/m)=Σ_(m≥1) (((−x)^m )/m)Σ_(n≥1) (1/n^(m+1) ) ∴ φ(x)=Σ_(m≥1) (((−x)^m ζ(m+1))/m) =(((−x)^1 )/1) ζ(2)+(x^2 /2)ζ(3)−(x^3 /3)ζ(4)+... ⇒((2φ(x))/x^2 )=ζ(3)−((2ζ(2))/x)−((2x)/3)ζ(4)+... ⇒2((φ(x))/x^2 )+(π^2 /(3x))=ζ(3)−((2x)/3)φζ(4)+... limit from both sides as x→0... lim_(x→0) (((2φ(x))/x^2 )+(π^2 /(3x)))=ζ(3) ✓✓ m.n.
solution:ϕ(x)=−∫01{tx−1ln(t)∑∞n=1tn−1n}dx=−∑∞n=11n∫01tx+n−1−tn−1ln(t)dt=−∑n⩾11n[ln(x+n)−ln(n)]=−∑n⩾11nln(x+nn)=−∑n⩾11nln(1+xn)=−∑n⩾11n∑m⩾1(−xn)mm=∑m⩾1(−x)mm∑n⩾11nm+1∴ϕ(x)=∑m⩾1(−x)mζ(m+1)m=(−x)11ζ(2)+x22ζ(3)−x33ζ(4)+…⇒2ϕ(x)x2=ζ(3)−2ζ(2)x−2x3ζ(4)+…⇒2ϕ(x)x2+π23x=ζ(3)−2x3ϕζ(4)+…limitfrombothsidesasx→0…limx→0(2ϕ(x)x2+π23x)=ζ(3)✓✓m.n.

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