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}

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