A Telescoping Log Product

★ 8/10functions Monotonicity sequences Logarithms Telescoping

Problem

Let $f(x) = \log_{x+1}(x+2)$ for $x \geqslant 1$ , and

g(x) = f(x)\cdot f(x+1)\cdots f(x+n) \qquad (n \in \mathbb{N}^*).

Let $F(n)$ be the maximum of $g(x)$ on $[1, +\infty)$ . Find the value of $n$ at which $|F(n+1) - F(n)|$ is largest.

Answer

Solution

Difficulty	8/10
Topics	functions, Monotonicity, sequences, Logarithms, Telescoping

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