Limit of a harmonic subseries minus "its" logarithm lim n → ∞ ∑ k =...
Limit of a harmonic subseries minus "its" logarithm
I think that inserting the other terms and then subtracting them would not help. I need just the ideea. Thank you.
Answer & Explanation
It is not very difficult to show that, for natural values of a, , where is the harmonic number. But in our case, , so we have to extend the formula or definition of so as to encompass non-natural arguments as well. Luckily for us, Euler has already done it some years ago: . Using this, we can now compute the value of to be . Now use the fact that . See here for more details on how to handle the integral in question.
If it were , we'd end up with . So I'd suggest to investigate