Does anyone know the exact value of this:

∞∑k=1(−1)kHkk

or this:

∞∑k=1(−1)kH(2)kk

Thanks!

Thanks again for the answers! I found very interesting that the integral gives exact values up to r=3 but from 4 this integral gives not exact values:

∫0−1Li4(t)t(1−t)dt

because wolfram says that “no results found in terms of standard mathematical functions”

**Answer**

From my answer in my previous avatar, denoting

A(p,q)=∞∑k=1(−1)k+1H(p)kkq

we have

A(1,1)=ζ(2)−log2(2)2

and

A(2,1)=ζ(3)−ζ(2)log(2)2

**Attribution***Source : Link , Question Author : Antal Korpa , Answer Author : Community*