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=∫101−yt1−y dy
If anyone has doubts about convergence, we have:
limt→0Htt=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
1−yt1−y=(1−yt)(1+y+y2+…)=(1−yt)+y(1−yt)+y2(1−yt)+…
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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