lect118_29_w13 - Wednesday March 20 text Lecture 29...

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Wednesday, March 20 ±Lecture 29 :Binomial series (Refers to Section 8.8 in your text) After having practiced the problems associated to the concepts of this lecture the student should be able to: State and apply the Binomial series theorem. 29.1Introduction±Recall the Binomial theorem, a theoremwhich most students have studied in high school: If nis a positive integer then where ܥ(݊,݇) =݊݇= ݊!(݊ െ ݇)!݇!We will study the Binomial series, a generalization of the expression (ܽ+ܾ). 29.1.1Definition ±Let be Dany real numberandkbe a non-negative integer. We define C(D, k)as follows: ܥ(ߙ, 0) =ߙ0= 1ܥ(ߙ,݇) =ߙ݇= Ƚ(Ƚ െ1)(Ƚ െ2)(Ƚ െ3) … (Ƚ െk + 1)݇!29.1.2Example ±By C(1/3, 5) we mean ܥ ൬13, 5=135= 13ቁ ቀ131ቁ ቀ132(133) …135 + 15!By C(±1/3, 5) we mean ܥ ൬െ13, 5=135= ቀെ13(131)ቀെ132ቁ ቀെ133ቀെ135 + 15!29.2Theorem ±The Binomial series theorem. Suppose Dis any real number. Then
with interval of convergence(±1, 1).

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