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Chapter2

Chapter2 - Chapter 2 Sequences and Series PWhite Discussion...

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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 [email protected] 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 Bolzano-Weierstrass 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 Bolzano-Weierstrass 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-...
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Chapter2 - Chapter 2 Sequences and Series PWhite Discussion...

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