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Unformatted text preview: MAT A26 Lecture 42 1 Alternating Series definition 1.1. A series is called alternating if it has the form ∑ ∞ k = k ( 1) k a k , (where k is some integer) example 1.2. 1 1 2 + 1 4 1 8 + ··· = ∑ ∞ k =0 ( 1) k 2 k is an alternating series. Note that a k = 1 2 k , 1 2 k > 1 2 k +1 > 0 for all k and lim k →∞ 1 2 k = 0. example 1.3. 1 1 2 + 1 3 1 4 + ··· = ∑ ∞ k =1 ( 1) k 1 k is an alternating series. Note that a k = 1 k , 1 k > 1 k +1 > 0 for all k and lim k →∞ 1 k = 0. proposition 1.4. The alternating series ∑ ( 1) n a n converges if: (i) a k ≥ a k +1 > 0 for all k , and (ii) lim k →∞ a k = 0. proof. Consider the even and odd partial sums s 2 n and s 2 n +1 separately. We have s 2 n +1 = s 2 n 1 +( a 2 n a 2 n +1 )  {z } > > s 2 n 1 and s 2 n +2 = s 2 n +( a 2 n +1 + a 2 n +2 )  {z } < < s 2 n Thus s 2 n +1 is monotone (increasing) and s 2 n is monotone (decreasing). More over s 2 n +1 = s 2 n a 2 n +1 > s 2 n In summary s 1 < s 3 < s 5 < ··· < s 6 < s 4 < s 2 So the sequence of odd partial sums has an upper bound (e.g., anySo the sequence of odd partial sums has an upper bound (e....
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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
 HIMONAS
 Calculus

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