Lecture42-rev

Lecture42-rev - MAT A26 Lecture 42 1 Alternating Series...

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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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Lecture42-rev - MAT A26 Lecture 42 1 Alternating Series...

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