AnalysisExam2

AnalysisExam2 - ( x n-x n +1 ) 2 = x 2 n +1-10 = x 2 n...

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Real Analysis: Exam II Extra Credit David Joseph Stith 3. Problem. Let ( x n ) be the sequence defned recursively by x 1 = 2 and x n +1 = x n 2 + 5 x n For all n 1 . Show that ( x n ) converges and fnd the limit oF the sequence. To show that ( x n ) converges, we will show that ( x n ) is bounded below and that ( x n ) is ultimately decreasing. It will then follow that ( x n ) is bounded above as well and hence, by the Monotone Convergence Theorem, converges. To show that ( x n ) is bounded below, we will show by Mathematical Induction that x n > 0 for all n N . We have that x 1 = 2 > 0. Suppose x k > 0 for some k N . Then x k / 2 > 0 and 5 /x k > 0 so that x k +1 = x k 2 + 5 x k > 0. Therefore x k > 0 = x k +1 > 0. Therefore by Mathematical Induction, x n > 0 for all n N . Therefore ( x n ) is bounded below. Furthermore we can easily see that x n R for all n N . Now we will show that x n +1 x n +2 for all n N and hence that the one-tail of ( x n ) is decreasing. Let n N . Then x n +1 = x n 2 + 5 x n = x n +1 = x 2 n + 10 2 x n = x 2 n - 2 x n x n +1 + 10 = 0 = x 2 n - 2 x n x n +1 = - 10 = x 2 n - 2 x n x n +1 + x 2 n +1 = - 10 + x 2 n +1 =
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Unformatted text preview: ( x n-x n +1 ) 2 = x 2 n +1-10 = x 2 n +1-10 = x 2 n +1 2 5 = x n +1 2 5 x n +1 since n N ,x n > = x n +1 x n +1 2 + 5 x n +1 = x n +1 x n +2 Therefore x n +1 x n +2 for all n N . Therefore x ( n ) is ultimately decreasing. Therefore < x n < x 2 for all n N so that ( x n ) is bounded as well as monotone. Therefore by the Monotone Convergence Theorem, ( x n ) is convergent. 1 To fnd lim( x n ), let x = lim( x n ) = lim( x n +1 ) by virtue oF the Fact that any tail oF ( x n ) also converges to x . Then, x = lim( x n +1 ) = lim( x n ) 2 + 5 lim( x n ) = x = x 2 + 5 x = 2 x 2 = x 2 + 10 = x 2 = 10 = x = 10 since n N ,x n > ThereFore lim( x n ) = 10. Q.E.D. 2...
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This note was uploaded on 03/05/2009 for the course MATH 117 taught by Professor Akhmedov,a during the Spring '08 term at UCSB.

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AnalysisExam2 - ( x n-x n +1 ) 2 = x 2 n +1-10 = x 2 n...

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