A Harmonic-Log Sandwich

★ 7/10calculus Monotonicity Logarithms Telescoping inequality

Problem

Let $f(x) = \dfrac{x}{1+x} - \ln(1+x)$ and $g(x) = \ln(1+x) - bx$ .

1. Find the equation of the tangent line to $y = f(x)$ at $(1, f(1))$ . 2. Is there a real number $b$ such that $g(x) < 0$ for all $x \in (0, +\infty)$ ? If so, find the range of such $b$ ; if not, explain why. 3. Prove that

\frac{1}{n} + \ln\frac{n}{e} < \sum_{k=1}^n \frac{k}{k^2 + 1} \leqslant \frac{1}{2} + \ln n.

Answer

Solution

Difficulty	7/10
Topics	calculus, Monotonicity, Logarithms, Telescoping, inequality

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