Odd Fractions to the $n$ -th Power

★ 8/10calculus Convexity sequences Estimation inequality

Problem

Let $f(x) = \mathrm{e}^x - x - 1$ .

1. Prove $f(x) \ge 0$ for all real $x$ . 2. Prove that for every positive integer $n$ ,

\left(\frac{1}{2n}\right)^n + \left(\frac{3}{2n}\right)^n + \left(\frac{5}{2n}\right)^n + \cdots + \left(\frac{2n-1}{2n}\right)^n < \frac{\sqrt{\mathrm{e}}}{\mathrm{e} - 1}.

Answer

Solution

Difficulty	8/10
Topics	calculus, Convexity, sequences, Estimation, 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