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lnln limit

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November 28, 2019
Sequenceslimitsreal analysis

Problem Statement

Let be a sequence (an)n≥1 \left( a_n \right)_{n\ge 1} with a1>0 a_1>0 and satisfying the equality an=an+1−an+1+an, a_n=\sqrt{a_{n+1} -\sqrt{a_{n+1} +a_n}} , for all natural numbers n. n.
a) Find a recurrence relation between two consecutive elements of (an)n≥1. \left( a_n \right)_{n\ge 1} . b) Prove that lim⁡n→∞ln⁡ln⁡ann=ln⁡2. \lim_{n\to\infty } \frac{\ln\ln a_n}{n} =\ln 2.
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