I have some difficulty to prove the following limit:
lim
Can someone help me? Thanks.
Answer
A common estimate for the Harmonic Numbers is
\sum_{k=1}^n\frac1k=\log(n)+\gamma+O\left(\frac1n\right)\tag{1}
where \gamma is the Euler-Mascheroni constant.
Applying (1), we get that
\begin{align} \sum_{k=1}^{N}\frac{1}{k+N} &=\sum_{k=1}^{2N}\frac1k-\sum_{k=1}^N\frac1k\\ &=\left(\log(2N)+\gamma+O\left(\frac{1}{2N}\right)\right)-\left(\log(N)+\gamma+O\left(\frac1N\right)\right)\\ &=\log(2)+O\left(\frac1N\right)\tag{2} \end{align}
Taking the limit of (2) as N\to\infty yields
\lim_{N\to\infty}\sum_{k=1}^{N}\frac{1}{k+N}=\log(2)\tag{3}
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