A Two-Sided Series Bound

★ 8/10Recursion sequences Estimation Telescoping inequality

Problem

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

na_{n+1} = (n+2)a_n + n,

and $b_n = \dfrac{a_n}{n(n+1)}$ .

1. Find $b_n$ and $a_n$ . 2. Prove that

\frac{n}{2} \leqslant \frac{1}{a_2} + \frac{1}{a_3} + \cdots + \frac{1}{a_{n+1}} + b_1^2 + b_2^2 + \cdots + b_n^2 \leqslant \frac{2n^2 - 4n + 4\sqrt{n} - 1}{2n}.

Answer

Solution

Difficulty	8/10
Topics	Recursion, sequences, Estimation, Telescoping, inequality

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