This preview shows pages 1–7. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Chapter 2: Sequences and Series PWhite Discussion The Limit of a Sequence The Algebraic and Order Limit Theorems MCT & Infinite Series Bolzano Weierstrass The Cauchy Criterion Properties of Infinite Series Double Sums & Products Epilogue Chapter 2: Sequences and Series Peter W. White white@tarleton.edu Initial development by Keith E. Emmert Department of Mathematics Tarleton State University Fall 2011 / Real Anaylsis I Chapter 2: Sequences and Series PWhite Discussion The Limit of a Sequence The Algebraic and Order Limit Theorems MCT & Infinite Series Bolzano Weierstrass The Cauchy Criterion Properties of Infinite Series Double Sums & Products Epilogue Overview Discussion: Rearrangements of Infinite Series The Limit of a Sequence The Algebraic and Order Limit Theorems The Monotone Convergence Theorem and a First Look at Infinite Series Subsequences and the BolzanoWeierstrass Theorem The Cauchy Criterion Properties of Infinite Series Double Summations and Products of Infinite Series Epilogue Chapter 2: Sequences and Series PWhite Discussion The Limit of a Sequence The Algebraic and Order Limit Theorems MCT & Infinite Series Bolzano Weierstrass The Cauchy Criterion Properties of Infinite Series Double Sums & Products Epilogue Example Example 1 Associativity need not hold when dealing with infinite series. Suppose that S = 1 1 2 + 1 3 1 4 + 1 5 = X n = 1 ( 1 ) n + 1 1 n . Then we can add half the sum to the original sum and obtain 1 2 S = 1 2 1 4 + 1 6 1 8 + + S = 1 1 2 + 1 3 1 4 + 1 5 1 6 + 1 7 1 8 3 2 S = 1 + 1 3 1 2 + 1 5 + 1 7 1 4 + Note that we have rearranged the original sum and Chapter 2: Sequences and Series PWhite Discussion The Limit of a Sequence The Algebraic and Order Limit Theorems MCT & Infinite Series Bolzano Weierstrass The Cauchy Criterion Properties of Infinite Series Double Sums & Products Epilogue Example Example 2 Another example: ( 1 + 1 ) + ( 1 + 1 ) + = + + = and moving parenthesis one step over, 1 + ( 1 1 ) + ( 1 1 ) + = 1 + + + = 1 . Remark 3 The conclusion: manipulations that are legal in the finite world need not extend to the infinite world...and we have yet to do something really creepy, like multiplying two infinite series! Chapter 2: Sequences and Series PWhite Discussion The Limit of a Sequence The Algebraic and Order Limit Theorems MCT & Infinite Series Bolzano Weierstrass The Cauchy Criterion Properties of Infinite Series Double Sums & Products Epilogue Overview Discussion: Rearrangements of Infinite Series The Limit of a Sequence The Algebraic and Order Limit Theorems The Monotone Convergence Theorem and a First Look at Infinite Series Subsequences and the BolzanoWeierstrass Theorem The Cauchy Criterion Properties of Infinite Series Double Summations and Products of Infinite Series Epilogue Chapter 2: Sequences and Series PWhite Discussion The Limit of a Sequence The Algebraic and Order Limit Theorems MCT & Infinite Series Bolzano...
View
Full
Document
This note was uploaded on 01/17/2012 for the course MATH 409 taught by Professor Dr.peterwhite during the Fall '11 term at Tarleton.
 Fall '11
 Dr.PeterWhite
 Algebra, Infinite Series, Sequences And Series

Click to edit the document details