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Unformatted text preview: MAT A26 Lecture 40 1 Series and Their Sums Definition 1. 1. A real (or complex) series is an expression of the form X k =0 a k where a k is a sequence of real (or complex) numbers. 2. The n th partial sum of the series is s n = n X k =0 a k (Note that s , s 1 , s 2 , s 3 , . . . is a sequence.) 3. We say the series converges to L and write k =0 a k = L if lim n s n = L (We say that L is the sum of the series. If the limit doesnt exist we say the series diverges or doesnt converge.) Example 1. (Geometric Series) Suppose that a 6 = 0 . Show that X k =0 ar k = a 1- r if | r | < 1 and that if | r | 1 then the series doesnt converge. SOLUTION: Consider the partial sum s n = n k =0 ar k . If r = 1 then s n = ( n + 1) a so lim n s n doesnt exist. So we may assume that r 6 = 1. Write s n = a + ar + ar 2 + + ar n- 1 + ar n rs n = ar + ar 2 + + ar n + ar n +1 s n- rs n = a- ar n +1 (1- r ) s n = a (1- r n +1 ) s n = a 1- r n +1 1- r since r 6 = 1 1 so lim n s n = a 1 1- r + a 1 1- r lim n r n...
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This note was uploaded on 03/04/2012 for the course MATH 10250 taught by Professor Himonas during the Fall '08 term at Notre Dame.
- Fall '08