Limit as a definite Integral

Question

Question: Express the given limit as a definite integral. Do not evaluate the Limit $$\lim_{n\to\infty} \sum_{i=1}^{n} \frac{1}{i+n}$$ My attemp: $$\int_{0}^{1}\frac{1}{x} dx$$ I think the limits of integration are correct but I am not really sure what function is being integrated. Is it correct ?

Answer 1

Note that $$ \lim_{n\to\infty} \sum_{i=1}^{n} \frac{1}{i+n}=\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n\frac{1}{1+\frac{i}{n}}=\int_{0}^1\frac{1}{1+x}\,dx $$ since we may recognize the second limit as limit of Riemann sums for $f(x)=\frac{1}{1+x}$ on $[0,1]$.