limit of a sequence

Question

How do I find the limit of a sequence: (1^n+2^n+…+n^n) /n^n, as n → ∞?

Answer 1

The sequence can be written as:

(1/n)^n + (2/n)^2 + (3/n)^n + .... + (n/n)^n =

(1/n)^n + (2/n)^2 + (3/n)^n + .... + (1)^n =

For infinitely large N, all but the last terms tend to zero.

the last term tends to 1.

So the limit is 1

Answer 2

Note that ln(1-x) < -x for x in [0,1], but lim x -> 0 ln(1-x)/x -> -1

Thus, lim n -> ∞ ((n-k)/n)ⁿ = e^-k , and the series increases to 1 + e^-1 + e^-2 + e^-3 ... has leading term 1 and ratio 1/e, so the sum is 1/(1 - 1/e) or e/(e-1)