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

approaches

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

approaches

Equation B is equivalent to

The Bernoulli polynomials 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:

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