A Harmonic Recurrence

★ 7/10sequences Estimation AM-GM Telescoping inequality

Problem

A sequence $\{a_n\}$ has $a_1 = 1$ and

a_{n+1} = \frac{(n+1)a_n}{n + 1 + a_n}.

Determine, with proof, which of the following hold for all applicable $n$ :

1. $\dfrac{1}{a_{n+1}} - \dfrac{1}{a_n} \geqslant \dfrac{1}{2}$ ; 2. $\dfrac{1}{a_{n+2}} - \dfrac{1}{a_n} < \dfrac{2}{\sqrt{(n+2)(n+1)}}$ ; 3. $\dfrac{1}{a_{2n}} - \dfrac{1}{a_n} \geqslant \dfrac{1}{2}$ ; 4. $a_n \ln(n+1) > 1$ .

Answer

Solution

Difficulty	7/10
Topics	sequences, Estimation, AM-GM, Telescoping, inequality

Whiteboard

Your sketch is saved only in this browser. To share it, export your drawing as an image (whiteboard menu → Export as → PNG), then upload that image in the comments below.

Discussion

Ask questions, share alternate solutions, and use LaTeX freely.

0 comments

No comments yet.