1-n-1-1-n-H-n-n-2-2-1- Tinku Tara June 4, 2023 Set Theory 0 Comments FacebookTweetPin Question Number 189091 by mnjuly1970 last updated on 12/Mar/23 1:Ω=∑∞n=1(−1)nHnn2=?2:η(−1)=? Answered by qaz last updated on 15/Mar/23 A=Σ(−1)nHn∫0∞xe−nxdx=−∫0∞xln(1+e−x)1+e−xdx=∫01lntln(1+t)t(1+t)dt=∫01(lntln(1+t)t−lntln(1+t)1+t)dt=∫01lnt(Li2(−t))′dt−∫01(lnt1+t+ln(1+t))ln(1+t)1+tdt=−∫01Li2(−t)tdt−13ln32−∫01ln(1−11+t)ln(1+t)1+tdt=Li3(−1)−13ln32−∫01(Li2(11+t))′ln(1+t)dt=Li3(−1)−13ln32−Li2(12)ln2+∫01Li2(11+t)1+tdt=Li3(−1)−13ln32−Li2(12)ln2−∫01(Li3(11+t))′dt=Li3(−1)−13ln32−Li2(12)ln2−Li3(12)+Li3(1)=Li3(−1)−13ln32−(π212−12ln22)ln2−[78Li3(1)−π212ln2+16ln32]+Li3(1)=Li3(−1)+18Li3(1)=(21−3−1)ζ(3)+18ζ(3)=−58ζ(3)… Commented by senestro last updated on 02/Apr/23 Terms of Service Privacy Policy Contact: info@tinkutara.com FacebookTweetPin Post navigation Previous Previous post: Question-189090Next Next post: Trace-the-changes-in-the-sign-and-magnitude-of-sin-3-cos-2-as-the-angle-increases-from-0-to-pi-2-also-find-its-minimum-and-maximum-values- Leave a Reply Cancel replyYour email address will not be published. Required fields are marked *Comment * Name * Save my name, email, and website in this browser for the next time I comment.
![A=Σ(−1)^n H_n ∫_0 ^∞ xe^(−nx) dx =−∫_0 ^∞ ((xln(1+e^(−x) ))/(1+e^(−x) ))dx=∫_0 ^1 ((lntln(1+t))/(t(1+t)))dt =∫_0 ^1 (((lntln(1+t))/t)−((lntln(1+t))/(1+t)))dt =∫_0 ^1 lnt(Li_2 (−t))′dt−∫_0 ^1 (((ln(t/(1+t))+ln(1+t))ln(1+t))/(1+t))dt =−∫_0 ^1 ((Li_2 (−t))/t)dt−(1/3)ln^3 2−∫_0 ^1 ((ln(1−(1/(1+t)))ln(1+t))/(1+t))dt =Li_3 (−1)−(1/3)ln^3 2−∫_0 ^1 (Li_2 ((1/(1+t))))′ln(1+t)dt =Li_3 (−1)−(1/3)ln^3 2−Li_2 ((1/2))ln2+∫_0 ^1 ((Li_2 ((1/(1+t))))/(1+t))dt =Li_3 (−1)−(1/3)ln^3 2−Li_2 ((1/2))ln2−∫_0 ^1 (Li_3 ((1/(1+t))))′dt =Li_3 (−1)−(1/3)ln^3 2−Li_2 ((1/2))ln2−Li_3 ((1/2))+Li_3 (1) =Li_3 (−1)−(1/3)ln^3 2−((π^2 /(12))−(1/2)ln^2 2)ln2−[(7/8)Li_3 (1)−(π^2 /(12))ln2+(1/6)ln^3 2]+Li_3 (1) =Li_3 (−1)+(1/8)Li_3 (1)=(2^(1−3) −1)ζ(3)+(1/8)ζ(3)=−(5/8)ζ(3) ...](https://www.tinkutara.com/question/Q189349.png)
