Find the limit $\lim_{n\rightarrow\infty}\sum_{k=1}^{n}\frac{1}{k+n}$

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  • The limit of truncated sums of harmonic series, $\lim\limits_{k\to\infty}\sum_{n=k+1}^{2k}{\frac{1}{n}}$

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Solutions Collecting From Web of "Find the limit $\lim_{n\rightarrow\infty}\sum_{k=1}^{n}\frac{1}{k+n}$"

$$\lim_{n\rightarrow\infty}\sum_{k=1}^{n}\frac{1}{k+n} =\lim_{n\rightarrow\infty}\dfrac1n\sum_{k=1}^{n}\frac1{\dfrac kn+1}$$

Use $$\lim_{n \to \infty} \frac1n\sum_{r=1}^n f\left(\frac rn\right)=\int_0^1f(x)dx$$

$$\sum_{k=1}^n\frac1{k+n}~=~\sum_{k=n+1}^{2n}\frac1k~=~\sum_{k=1}^{2n}\frac1k~-~\sum_{k=1}^n\frac1k~=~H_{2n}-H_n~\simeq~\ln2n-\ln n~=~\ln\frac{2n}n$$ Can you take it from here, and evaluate what happens as $n\to\infty$ ? :-$)$