Evaluate limit of the series: $lim_{ntoinfty} left[frac{1}{n^{2}} + frac{2}{n^{2}} + frac{3}{n^{2}} + cdots + frac{n}{n^{2}}right]$

I am pretty confident that the following limit is $0.5$:

$$\lim_{n\to\infty} \left[\frac{1}{n^{2}} + \frac{2}{n^{2}} + \frac{3}{n^{2}} + \cdots + \frac{n}{n^{2}}\right]=\lim_{n\to\infty} \left[\frac{1+2+3+ \cdots +n}{n^{2}}\right]=\lim_{n\to\infty} \left[\frac{n^2+n}{2n^{2}}\right]=\frac{1}{2}$$

However one of the students argued that if we write limit of sum as sum of individual limits, it will be zero. Why we cannot write limit of sum as sum of limits in this case?

I've been taught that if individual limits exist, the limit of sum is equal to the sum of limits. It would be helpful to get an explanation or a reference to similar rules for limits of series.

Answer

However one of the students argued that if we write limit of sum as sum of individual limits, it will be zero. Why we cannot write limit of sum as sum of limits in this case?

For the sum of two sequencesn we have the property "limit of the sum is the sum of limits" (if both sequences have a limit) and by repeatedly applying this, we have this property for any finite number of sequences (terms).

As often is the case, you cannot simply extend this to the infinite case; i.e. you cannot assume the same property will hold when the number of sequences (terms) is not finite.

A simpler counterexample would be the sum of $n$ terms, all equal to $\tfrac{1}{n}$; obviously we have:
$$\underbrace{\frac{1}{n}+\frac{1}{n}+\ldots+\frac{1}{n}}_{\mbox{$n$ terms}} = \frac{n}{n}=1$$
but every individual sequence (term) clearly tends to $0$: $\frac{1}{n} \stackrel{n\to \infty}{\longrightarrow} 0$.