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Chap11_Sec2 - 11 INFINITE SEQUENCES AND SERIES INFINITE...

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11 INFINITE SEQUENCES AND SERIES INFINITE SEQUENCES AND SERIES
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11.2 Series In this section, we will learn about: Various types of series. INFINITE SEQUENCES AND SERIES
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SERIES If we try to add the terms of an infinite sequence we get an expression of the form a 1 + a 2 + a 3 + ··· + a n + ··· 1 { } n n a = Series 1
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INFINITE SERIES This is called an infinite series (or just a series). It is denoted, for short, by the symbol 1 or n n n a a =
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However, does it make sense to talk about the sum of infinitely many terms? INFINITE SERIES
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It would be impossible to find a finite sum for the series 1 + 2 + 3 + 4 + 5 + ··· + n + ··· If we start adding the terms, we get the cumulative sums 1, 3, 6, 10, 15, 21, . . . After the n th term, we get n ( n + 1)/2, which becomes very large as n increases. INFINITE SERIES
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However, if we start to add the terms of the series we get: 1 1 1 1 1 1 1 2 4 8 16 32 64 2 n + + + + + + + ××× + ××× 3 7 15 31 63 1 2 4 8 16 32 64 , , , , , , ,1 1/ 2 , n ×× - × ××× INFINITE SERIES
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The table shows that, as we add more and more terms, these partial sums become closer and closer to 1. In fact, by adding sufficiently many terms of the series, we can make the partial sums as close as we like to 1. INFINITE SERIES
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So, it seems reasonable to say that the sum of this infinite series is 1 and to write: 1 1 1 1 1 1 1 1 2 2 4 8 16 2 n n n = = + + + + + + ××× ××× = INFINITE SERIES
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We use a similar idea to determine whether or not a general series (Series 1) has a sum. INFINITE SERIES
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We consider the partial sums s 1 = a 1 s 2 = a 1 + a 2 s 3 = a 1 + a 2 + a 3 s 3 = a 1 + a 2 + a 3 + a 4 In general, 1 2 3 1 n n n i i s a a a a a = ×× + = × = + + + INFINITE SERIES
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These partial sums form a new sequence { s n }, which may or may not have a limit. INFINITE SERIES
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If exists (as a finite number), then, as in the preceding example, we call it the sum of the infinite series Σ a n . lim n n s s →∞ = SUM OF INFINITE SERIES
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Given a series let s n denote its n th partial sum: 1 2 3 1 n n a a a a = = + + +××× 1 2 1 n n i n i s a a a a = = = + +××× + SUM OF INFINITE SERIES Definition 2
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If the sequence { s n } is convergent and exists as a real number, then the series Σ a n is called convergent and we write: The number s is called the sum of the series. Otherwise, the series is called divergent. 1 2 1 or n n n a a a s a s = + +××× + +×××= = SUM OF INFINITE SERIES Definition 2 lim n n s s →∞ =
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Thus, the sum of a series is the limit of the sequence of partial sums. So, when we write , we mean that, by adding sufficiently many terms of the series, we can get as close as we like to the number s . 1 n n a s = = SUM OF INFINITE SERIES
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Notice that: 1 1 lim n n i n n i a a →∞ = = = SUM OF INFINITE SERIES
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Compare with the improper integral To find this integral, we integrate from 1 to t and then let t → ∞.
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