<s>[INST] What is $\lim_{n \rightarrow \infty}\sum\limits_{k = 1}^n\frac{k}{n^2}$? We have $$\dfrac{1+2+3+...+ \space n}{n^2}$$
What is the limit of this function as $n \rightarrow \infty$?
My idea:
$$\dfrac{1+2+3+...+ \space n}{n^2} = \dfrac{1}{n^2} + \dfrac{2}{n^2} + ... + \dfrac{n}{n^2} = 0$$
Is this correct?
 [/INST] HINT:
Such application of limit (esp. to $\infty$) to individual summand is applicable only when the number of summand is finite.
$$\dfrac{1+2+3+...+ \space n}{n^2} =\frac{\frac{n(n+1)}2}{n^2}=\frac12\cdot\left(1+\frac1n\right)$$
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