A Symmetric Pairing Sum

★ 7/10functions sequences Estimation Symmetry

Problem

Define $f(x) = \dfrac{4^x}{4^x + 2}$ on $\mathbb{R}$ , and for $n = 2, 3, \ldots$ set

S_n = f\left(\frac{1}{n}\right) + f\left(\frac{2}{n}\right) + \cdots + f\left(\frac{n-1}{n}\right).

1. Find $S_n$ . 2. Is there a constant $M > 0$ such that for all $n \geqslant 2$ ,

\frac{1}{S_2} + \frac{1}{S_3} + \cdots + \frac{1}{S_{n+1}} \leqslant M\,?

Justify your answer.

Answer

Solution

Difficulty	7/10
Topics	functions, sequences, Estimation, Symmetry

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