Question

How do you find the limit of `s(n)=81/n^4[(n^2(n+1)^2)/4]` as `n->oo`?

Answer 1

`:lim_(n rarr oo) s(n) = 81/4`

Explanation:

`s(n) = 81/n^4[(n^2(n+1)^2)/4]`
`:. s(n) = 81/n^2[((n+1)^2)/4]`
`:. s(n) = 81/n^2[(n^2+2n+1)/4]`
`:. s(n) = 81/4[(n^2+2n+1)/n^2]`
`:. s(n) = 81/4[n^2/n^2+(2n)/n^2+1/n^2]`
`:. s(n) = 81/4[1+2/n+1/n^2]`

So then,
`lim_(n rarr oo) s(n) = lim_(n rarr oo) 81/4[1+2/n+1/n^2]`
`:. lim_(n rarr oo) s(n) = 81/4` as both `2/n rarr 0`, and `1/n^2 rarr 0` as `n rarr oo`