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## Homework Statement

Find [itex]\stackrel{lim}{_{n\rightarrow\infty}}\frac{1^{1}+2^{2}+...+(n-1)^{n-1}+n^{n}}{n^{n}}[/itex].

**2. The attempt at a solution**

At first I split up the fraction into a sum of a bunch of terms, and said that all of the terms went to 0 except the last, which is 1. But then I realized that in the limit, there is an infinite number of terms so it makes no sense to say that the "last term" is 1. I think I found a way to prove it by finding a recursive formula for the function inside the limit and then using that to get upper and lower bounds, which both converge to 1. But that was really long and seemed like a very round-about way of doing things. Is there a simpler way to get the answer?