Asymptotic behaviour of a polynomial root

[nugae]

I've been looking at the value N(n) of N that satisfies the equation

$$\sum_{1}^{n}(N-i)^{n}=N^{n}$$

Thus turns out to be

$$N(n)=1.5+\frac{n}{ln2}+O(1/n)$$

where the O(1/n) term is about 1/400n for n>10.

I've verified this by calculation up to about n=1000, using Lenstra's long integer package LIP.

This result is so beautiful and simple that it must be possible to prove it without brute-force calculation. If anyone has any suggestions as to how to begin then I'd be very grateful!

 

[Werg22]

Hummm looking at it, I'd say that if you examine the expansion of (N-i)^n and then study how do things look when you add from 1 to n, there should be a series that emerges which can be simplified to a fundamental function. However, this could be misleading as the expansion of these functions have an infinite number of terms and clearly this isn't the case on the left.

 

[Hurkyl]

Well, usually these sorts of things don't turn out to be equal to an elementary function -- you have to settle for approximately equal.

The problem is, I don't yet see any useful approximation to that series, or even to part of it.