Find limit by summation of series or by using Cauchy's first limit theorem

Question

I have to find the limit of this problem I have found that problem like this I always get two answers one by using the Cauchy first limit theorem and second by definite integral method so I am confused as to use which method can I not apply Cauchy theorem on this problem are there some conditions that I am overlooking.

This is a in general problem not specific problem. Please explain this to me

Answer 1

The theorem you're referring to says : if $x_n\to l$ then $y_n=\sum_i^nx_i/n\to l$. Now if $x_n=n^2/(n^2+n^2)$ then

$$y_n={\sum_{i=1}{i^2\over i^2+i^2}\over n}\neq {\sum_{i=1}^n{n^2\over n^2+i^2}\over n}$$

So you're just confused with the indices