Keeping a Recurrence Below 1

★ 8/10Recursion sequences Estimation Telescoping inequality

Problem

A sequence has $a_0 = \dfrac1k$ ( $k$ a positive integer) and

a_n = a_{n-1} + \frac{1}{n^2}a_{n-1}^2.

Find all $k$ such that $a_n < 1$ for every $n \in \mathbb{N}^*$ .

Answer

Solution

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

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