Question

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

Answer 1

`lim_(n rarr oo) s(n) = 1/2`

Explanation:

We have;
`s(n) = 1/n^2[(n(n+1))/2]`
`:. s(n) = 1/2[(n^2+n)/(n^2)]`
`:. s(n) = 1/2[1+1/n]`
`:. s(n) = 1/2 + 1/(2n)`

`lim_(n rarr oo) s(n) = lim_(n rarr oo){ 1/2 + 1/(2n) }`

Now `1/(2n) rarr 0` as `n rarr oo`, and so;
`lim_(n rarr oo) s(n) = 1/2+0 = 1/2`