Chap11_Sec5 - 11 INFINITE SEQUENCES AND SERIES INFINITE...

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11 INFINITE SEQUENCES AND SERIES INFINITE SEQUENCES AND SERIES
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INFINITE SEQUENCES AND SERIES The convergence tests that we have looked at so far apply only to series with positive terms.
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11.5 Alternating Series In this section, we will learn: How to deal with series whose terms alternate in sign. INFINITE SEQUENCES AND SERIES
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ALTERNATING SERIES An alternating series is a series whose terms are alternately positive and negative. Here are two examples: 1 1 1 1 1 1 1 1 ( 1) 1 ... 2 3 4 5 6 1 2 3 4 5 6 ... ( 1) 2 3 4 5 6 7 1 n n n n n n n - = = - - + - + - + = - + - + - + - = = +
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ALTERNATING SERIES From these examples, we see that the n th term of an alternating series is of the form a n = (–1) n – 1 b n or a n = (–1) n b n where b n is a positive number. In fact, b n = | a n |
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ALTERNATING SERIES The following test states that, if the terms of an alternating series decrease toward 0 in absolute value, the series converges.
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ALTERNATING SERIES TEST If the alternating series satisfies i. b n +1 b n for all n ii. then the series is convergent. 1 1 2 3 4 5 6 1 ( 1) ... 0 n n n n b b b b b b b b - = - = - + - + - + lim 0 n n b →∞ =
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ALTERNATING SERIES Before giving the proof, let’s look at this figure —which gives a picture of the idea behind the proof.
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ALTERNATING SERIES First, we plot s 1 = b 1 on a number line. To find s 2 ,we subtract b 2. So, s 2 is to the left of s 1 .
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ALTERNATING SERIES Then, to find s 3 , we add b 3. So, s 3 is to the right of s 2 . However, since b 3 < b 2 , s 3 is to the left of s 1 .
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ALTERNATING SERIES Continuing in this manner, we see that the partial sums oscillate back and forth. Since b n → 0, the successive steps are becoming smaller and smaller.
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The even partial sums s 2 , s 4 , s 6 , . . . are increasing.
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This note was uploaded on 01/06/2012 for the course MATH 2414.S01 taught by Professor Alans.grave during the Fall '11 term at Collins.

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Chap11_Sec5 - 11 INFINITE SEQUENCES AND SERIES INFINITE...

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