(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Finding the limit of a complicated funcion

**Physics Forums | Science Articles, Homework Help, Discussion**