Asymptotic behaviour of a polynomial root

nugae

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

[tex] \sum_{1}^{n}(N-i)^{n}=N^{n} [/tex]

Thus turns out to be

[tex] N(n)=1.5+\frac{n}{ln2}+O(1/n) [/tex]

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.