Tuesday, 25 October 2016

integration - Prove the integral int10fracHttdt=suminftyk=1fracln(1+frac1k)k



By numerical results it follows that:



10Httdt=k=1ln(1+1k)k=1.25774688694436963



Here Ht is the harmonic number, which is the generalization of harmonic sum and has an integral representation:




Ht=101yt1y dy






If anyone has doubts about convergence, we have:



limt0Htt=π26



Which would be another nice thing to prove, although I'm sure this proof is not hard to find.







It is also interesting that the related integral gives Euler-Mascheroni constant:



10Htdt=γ


Answer



We apply
1yt1y=(1yt)(1+y+y2+)=(1yt)+y(1yt)+y2(1yt)+
Each term gives after y-integration,
tt+1+t2(t+2)+t3(t+3)+
Then we divide these by t,
1t+1+12(t+2)+13(t+3)+
Taking integral with t variable, we have the result.




Any interchange of integral and summation can be justified by Monotone Convergence Theorem.


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