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Unformatted text preview: 11 INFINITE SEQUENCES AND SERIES INFINITE SEQUENCES AND SERIES 11.2 Series In this section, we will learn about: Various types of series. INFINITE SEQUENCES AND SERIES 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 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 = However, does it make sense to talk about the sum of infinitely many terms? INFINITE SERIES 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 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 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 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 We use a similar idea to determine whether or not a general series (Series 1) has a sum. INFINITE SERIES 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 These partial sums form a new sequence { s n }, which may or may not have a limit. INFINITE SERIES If exists (as a finite number), then, as in the preceding example, we call it the sum of the infinite series a n . lim n n s s = SUM OF INFINITE SERIES 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 If the sequence { s n } is convergent and exists as a real number, then the series 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 = 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 ....
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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.
 Fall '11
 AlanS.Grave
 Harmonic Series, Sequences And Series

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