$\lim\limits_{n \to \infty }\sum_{i=1}^{n}{1 \over n+i-1} = \int_{1}^{2}{1 \over x}dx$

Question

$\lim\limits_{n \to \infty }\sum_{i=1}^{n}{1 \over n+i-1} = \int_{1}^{2}{1 \over x}dx$

For the above equations, could one can prove it without using any derivatives?

Answer 1

We can write this as a Riemann Sum by dividing by $n$ and switch indices to get $$\frac{1}{n}\sum_{i=1}^n \frac{1}{1+\frac{i-1}{n}}=\frac{1}{n}\sum_{i=2}^n \frac{1}{1+\frac{i}{n}}$$ Convert this to a Riemann Integral and use properties of integrals to get your form