How do you find the limit of s(n)=1/n^2[(n(n+1))/2] as n->oo?

Nov 7, 2016

${\lim}_{n \rightarrow \infty} s \left(n\right) = \frac{1}{2}$

Explanation:

We have;
$s \left(n\right) = \frac{1}{n} ^ 2 \left[\frac{n \left(n + 1\right)}{2}\right]$
$\therefore s \left(n\right) = \frac{1}{2} \left[\frac{{n}^{2} + n}{{n}^{2}}\right]$
$\therefore s \left(n\right) = \frac{1}{2} \left[1 + \frac{1}{n}\right]$
$\therefore s \left(n\right) = \frac{1}{2} + \frac{1}{2 n}$

${\lim}_{n \rightarrow \infty} s \left(n\right) = {\lim}_{n \rightarrow \infty} \left\{\frac{1}{2} + \frac{1}{2 n}\right\}$

Now $\frac{1}{2 n} \rightarrow 0$ as $n \rightarrow \infty$, and so;
${\lim}_{n \rightarrow \infty} s \left(n\right) = \frac{1}{2} + 0 = \frac{1}{2}$