# A mathematical problem

This problem is in two parts. Part A is A-level standard; part B is more advanced.

A. Given that k > 1.0, prove that, for large n, the largest real root of

$(x+1)^{n}=kx^{n}$

approaches

$x\sim\frac{n}{\ln k}-\frac{1}{2}$

B. Given that k > 1.0, prove that, for large n, the largest real root of

$B_{n+1}(x+1)=kB_{n+1}(x)$

approaches

$x\sim\frac{n}{\ln k}+\frac{1}{k-1}+\frac{1}{2}$

Equation B is equivalent to

$(n+1)x^{n}=(k-1)B_{n+1}(x)$

The Bernoulli polynomials $B_{n}(x)$ are described in MathWorld and Wikipedia. Numbering schemes differ: this problem uses what MathWorld describes as the “new” definition, with the first few polynomials being:

$B_{0}(x)=1$

$B_{1}(x)=x-\frac{1}{2}$

$B_{2}(x)=x^{2}-x+\frac{1}{6}$