continued-fractions - CONTINUED FRACTIONS ROMAN VERSHYNIN,...

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CONTINUED FRACTIONS ROMAN VERSHYNIN, MATH 451, FALL 2011 As an application of the theory of limits, we will verify that (1) 1 + 1 2 + 1 2 + 1 2 + ··· = 2 . An object in the left side of the equation is called a continued fraction . Equation (1) is one example of a rich theory of continuous fractions, see Wikipedia if interested. One can rigorously define the continuous fraction in (1) as a limit of the sequence of finite fractions of the form 1 , 1 + 1 2 , 1 + 1 2 + 1 2 , 1 + 1 2 + 1 2 + 1 2 ,... The first six terms of this sequence are approximately 1 , 1 . 5 , 1 . 4 , 1 . 417 , 1 . 4138 , 1 . 41429 , so the sequence indeed seems to converge to 2 1 . 414214 quite fast. The following theorem is a rigorous way to state this convergence. Theorem 1. Let a 1 = 1 and (2) a n +1 = 1 + 1 1 + a n , n = 1 , 2 ,.... Then lim a n = 2 . We shall prove the theorem by the following series of results. Lemma 2.
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This note was uploaded on 12/13/2011 for the course MATH 451 taught by Professor Staff during the Fall '08 term at University of Michigan.

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continued-fractions - CONTINUED FRACTIONS ROMAN VERSHYNIN,...

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