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Lecture40-rev

# Lecture40-rev - MAT A26 Lecture 40 1 Series and Their Sums...

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MAT A26 Lecture 40 1 Series and Their Sums Definition 1. 1. A real (or complex) series is an expression of the form 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 k =0 a k (Note that s 0 , 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 doesn’t exist we say the series diverges or doesn’t converge.) Example 1. (Geometric Series) Suppose that a = 0 . Show that k =0 ar k = a 1 - r if | r | < 1 and that if | r | ≥ 1 then the series doesn’t 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 doesn’t exist. So we may assume that r = 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 = 1 1

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so lim n →∞ s n = a 1 1 - r + a 1 1 - r lim n →∞ r n +1 Since lim n →∞ r n +1 = 0 if | r | < 1 and doesn’t exist if either | r | > 1 or r = -
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Lecture40-rev - MAT A26 Lecture 40 1 Series and Their Sums...

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