Packing Harmonic Squares

★ 8/10combinatorics Estimation inequality

Problem

Prove the following.

1. For every integer $k\geqslant 2$ ,

\frac 1{2^k}+\frac 1{2^k+1}+\frac 1{2^k+2}+\dots+\frac 1{2^{k+1}-1}<1.

The squares with side lengths $1,\dfrac 12,\dfrac 13,\dots,\dfrac 1n,\dots$ can all be placed, without overlap, inside a single square of side length $\dfrac 32$ .

Answer

Solution

Difficulty	8/10
Topics	combinatorics, Estimation, inequality

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