Reduce this lim, too many n's

(n+1) n^(n) / (n+1) ^(n+1)

this is my ratio test, I thought about dividing each term by n but I'm not sure if that would even help..thanks :)

(n+1)(n^n)/(n+1)*(n+1)^n = (n/(n+1))^n = 1/(1+1/n)^n
the limit is 1/e

(n+1)/(n+1)^(n+1) = 1/(n+1)^n. Think of it as a/a^(n+1) if you have trouble seeing that.

So that reduces it to n^n/(n+1)^n = (n/n+1)^n

Unfortunately I believe the limit of that expression as n->infinity is 1, meaning your ratio test is inconclusive.

(n + 1) * n^n / (n + 1)^(n + 1)
(n + 1) * n^n / ((n + 1) * (n + 1)^n)
n^n / (n + 1)^n
(n / (n + 1))^n


let n + 1 = u,

n + 1 = u
n = u - 1

(n / (n + 1))^n =>
((u - 1) / u)^(u - 1) =>
(1 + (-1)/u)^(u) / (1 + (-1)/u)

u goes to infinity, since inf = u - 1 =>> u = inf

Remember the definition of e^x =>> (1 + x/t)^(t), t goes to infinity

(1 + (-1)/u)^u =>> e^(-1) => 1/e

(1/e) / (1 + -1/inf) =>
(1/e) / (1 + 0) =>
(1/e) / 1 =>
1/e

The limit is 1/e