Riemann sum computation

Question

I'm trying to evaluate $\int_1^2\frac{1}{x^2}dx$ using Rieman sums. I subdivide $[1,2]$ with $1=x_0<x_1<...<x_n=2$ and choose $\theta_i=\sqrt{x_{i-1}x_i}$ inside $[x_{i-1},x_i]$. I thus get the sum

$$\sum_{k=0}^n\frac{x_i-x_{i-1}}{x_{i-1}x_i}$$

but i'm not able to change this expression to compute its limit and get the value of $\int_1^2\frac{1}{x^2}dx$ (without going through primitive). Any clue?

Answer 1

Use $$ \frac{a-b}{ab}=\frac1b-\frac1a $$ to obtain a telescoping sum.