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Prove that the limit is 2 - [Iran Second Round 1989]

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December 6, 2010
limitinductionalgebra proposedalgebra

Problem Statement

Let {an}n≥1\{a_n\}_{n \geq 1} be a sequence in which a1=1a_1=1 and a2=2a_2=2 and an+1=1+a1a2a3⋯an−1+(a1a2a3⋯an−1)2∀n≥2.a_{n+1}=1+a_1a_2a_3 \cdots a_{n-1}+(a_1a_2a_3 \cdots a_{n-1} )^2 \qquad \forall n \geq 2. Prove that lim⁡n→∞(1a1+1a2+1a3+⋯+1an)=2\lim_{n \to \infty} \biggl( \frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}+\cdots + \frac{1}{a_n} \biggr) =2
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