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1980 VTRMC #3

Source:

August 11, 2020
limitSequence

Problem Statement

Let an=1⋅3⋅5⋅⋯⋅(2n−1)2⋅4⋅6⋅⋯⋅2n.a_n = \frac{1\cdot3\cdot5\cdot\cdots\cdot(2n-1)}{2\cdot4\cdot6\cdot\cdots\cdot2n}. (a) Prove that lim⁡n→∞an\lim_{n\to \infty}a_n exists. (b) Show that an=(1−122)(1−142)(1−162)⋯(1−1(2n)2)(2n+1)an.a_n = \frac{\left(1-\frac1{2^2}\right)\left(1-\frac1{4^2}\right)\left(1-\frac1{6^2}\right)\cdots\left(1-\frac{1}{(2n)^2}\right)}{(2n+1)a_n}. (c) Find lim⁡n→∞an\lim_{n\to\infty}a_n and justify your answer
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