121616949-math.248 - are the real numbers or a subset of...

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234 Chapter 10 Sequences and Series and then lim i →∞ 1 - 1 2 i = 1 - 0 = 1 . There is one place that you have long accepted this notion of infinite sum without really thinking of it as a sum: 0 . 3333 ¯ 3 = 3 10 + 3 100 + 3 1000 + 3 10000 + · · · = 1 3 , for example, or 3 . 14159 . . . = 3 + 1 10 + 4 100 + 1 1000 + 5 10000 + 9 100000 + · · · = π. Our first task, then, to investigate infinite sums, called series , is to investigate limits of sequences of numbers. That is, we officially call X i =1 1 2 i = 1 2 + 1 4 + 1 8 + 1 16 + · · · + 1 2 i + · · · a series, while 1 2 , 3 4 , 7 8 , 15 16 , . . . , 2 i - 1 2 i , . . . is a sequence, and X i =1 1 2 i = lim i →∞ 2 i - 1 2 i , that is, the value of a series is the limit of a particular sequence. While the idea of a sequence of numbers, a 1 , a 2 , a 3 , . . . is straightforward, it is useful to think of a sequence as a function. We have up until now dealt with functions whose domains
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Unformatted text preview: are the real numbers, or a subset of the real numbers, like f ( x ) = sin x . A sequence is a function with domain the natural numbers N = { 1 , 2 , 3 , . . . } or the non-negative integers, Z ≥ = { , 1 , 2 , 3 , . . . } . The range of the function is still allowed to be the real numbers; in symbols, we say that a sequence is a function f : N → R . Sequences are written in a few different ways, all equivalent; these all mean the same thing: a 1 , a 2 , a 3 , . . . { a n } ∞ n =1 { f ( n ) } ∞ n =1 { a n | n ∈ N } { f ( n ) | n ∈ N }...
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