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two sequence

Source: 31-th Vietnamese Mathematical Olympiad 1993

February 17, 2007
invarianttrigonometryreal analysisreal analysis unsolved

Problem Statement

Define the sequences a0,a1,a2,...a_{0}, a_{1}, a_{2}, ... and b0,b1,b2,...b_{0}, b_{1}, b_{2}, ... by a0=2,b0=1,an+1=2anbn/(an+bn),bn+1=an+1bna_{0}= 2, b_{0}= 1, a_{n+1}= 2a_{n}b_{n}/(a_{n}+b_{n}), b_{n+1}= \sqrt{a_{n+1}b_{n}}. Show that the two sequences converge to the same limit, and find the limit.
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