sequences and series - Show $lim_{Nto infty}sum_{k=1}^{N}frac{1}{k+N}=ln(2)$

I have some difficulty to prove the following limit:

$$\lim_{N\to \infty}\sum_{k=1}^{N}\frac{1}{k+N}=\ln(2)$$
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
$$\lim_{N\to\infty}\sum_{k=1}^{N}\frac{1}{k+N}=\log(2)\tag{3}$$