Question

How do you determine whether the sequence `a_n=(n+1)^n/n^(n+1)` converges, if so how do you find the limit?

Answer 1

`lim_(n->oo) a_n = 0`

Explanation:

Note that:

`(1+1/n)^n`

is monotonically increasing with limit `e` as `n->oo`

So:

`lim_(n->oo) a_n = lim_(n->oo) (n+1)^n/(n^(n+1))`

`color(white)(lim_(n->oo) a_n) = lim_(n->oo) ((n+1)/n)^n/n`

`color(white)(lim_(n->oo) a_n) = lim_(n->oo) (1+1/n)^n/n`

`color(white)(lim_(n->oo) a_n) <= lim_(n->oo) e/n`

`color(white)(lim_(n->oo) a_n) = 0`