A Contracting Iteration

★ 7/10calculusMonotonicitysequencesEstimationInduction

Problem

A sequence {an}\{a_n\} satisfies a1=−12a_1 = -\dfrac{1}{2} and

an+1=ln⁡(an+1)−sin⁡an.a_{n+1} = \ln(a_n + 1) - \sin a_n.

Determine, with proof, which of the following hold:

1. an>an+1a_n > a_{n+1}; 2. −12⩽an⩽−14-\dfrac{1}{2} \leqslant a_n \leqslant -\dfrac{1}{4}; 3. an+1>−an2an+2a_{n+1} > -\dfrac{a_n^2}{a_n + 2}; 4. an⩾−12na_n \geqslant -\dfrac{1}{2^n}.

Answer

Solution

Difficulty7/10
Topicscalculus, Monotonicity, sequences, Estimation, Induction

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