This is a problem taken from an exam of Analysis **1** course I attend.

$$\lim\_{n\to \infty}\left(\frac{1}{5n-2}+\frac{1}{5n-1}+\cdots+\frac{1}{8n+2}\right)$$

I tryed to solve this limit and I've got the following results

$$\underbrace{\lim\_{n\to \infty}\left(\frac{3n+5}{5n-2}\right)}\_{{\longrightarrow}\frac{3}{5}}\geq\lim\_{n\to \infty}\sum\_{k=1}^{3n+5}\frac{1}{5n+k-3}\geq\underbrace{\lim\_{n\to \infty}\left(\frac{3n+5}{8n+2}\right)}\_{{\longrightarrow}\frac{3}{8}}$$

I was going for Squeeze theorem, but I got that $\text{LHS}\left(\frac{3}{5}\right)\space\ne\space \text{RHS}\left(\frac{3}{8}\right)$. Since the squeeze theorem works only in one direction, it's not guaranteed it diverges. 

*So, what did I do wrong or what other method to use on this problem ?*

Reminder: The limit is supposed to be solved only with the knowledge prior to derivatives and integrals.

EDIT: Are these kinds of problems called truncated sums or series, similar problem [here](https://math.stackexchange.com/questions/3521700/evaluate-lim-n-to-infty-frac1-sqrt4n4n2-cdots-frac1-sqrt)?

Thanks in advance