Theorem 7 ratio test let a n and b n series with

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Theorem 7. (Ratio Test) Let a n and b n series with positive terms such that a n +1 a n 6 b n +1 b n for any n , then If b n is convergent, then a n is convergent. If a n is divergent, then b n is divergent. 8
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Introductory Real Analysis Math 327, Summer 2015 University of Washington c 2015, Dr. F. Dos Reis Theorem 8. Let a n be a sequence with positive terms. If lim n →∞ u n +1 u n = r < 1 then u n is convergent. If lim n →∞ u n +1 u n = r > 1 then u n is divergent. If lim n →∞ u n +1 u n = 1, then the theorem does not give any conclu- sion. Theorem 9. (Comparison to a geometric series) If for any n , u n +1 u n < r < 1, then u n is convergent. Theorem 10. (comparison with an integral) Given a positive function f that is non increasing on a interval [1 , ). Then the series f ( n ) and Z 1 f ( t )d t are both convergent or they are both divergent. 2.2 Absolute convergence and conditional convergence Definition 11. Given a series u n , u n is absolutely convergent if | u n | is convergent. Theorem 12. If u n is absolutely convergent, then u n is con- vergent. Theorem 13. Given a series u n , if a n is the subsequence of u n corresponding to the positive terms.If b n is the subsequence of u n corresponding to the non positive terms ( possibly completed by 0 terms is one of the sequence is finite ) If u n is absolutely convergent, then a n and b n are convergent and u n = a n + b n . If u n is conditionally convergent, then both a n and b n are divergent. Definition 14. If u n is convergent and not absolutely conver- gent, then u n is conditionally convergent. Theorem 15. (Alternating series) A series u n such that its terms alternate between positive and negative, and such that | u n | is decreasing toward 0 is convergent. 9
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Introductory Real Analysis Math 327, Summer 2015 University of Washington c 2015, Dr. F. Dos Reis 3 Chapter 20 3.1 Pointwise convergence Definition 16. Given a sequence of functions f n , the sequence f n is pointwise convergent to f on an interval I if for any x in I , lim n →∞ f ( x ) = f ( x ).
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  • Winter '08
  • Staff
  • Math, Calculus, Mathematical analysis, Modes of convergence, Introductory Real Analysis, Dr. F. Dos Reis

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