cal10 - Chapter Ten Sequences, Series, and All That 10.1...

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10.1 Chapter Ten Sequences, Series, and All That 10.1 Introduction Suppose we want to compute an approximation of the number e by using the Taylor polynomial p n for f x e x ( ) = at a =0. This polynomial is easily seen to be p x x x x x n n n ( ) ! = + + + + + 1 2 6 2 3 K . We could now use p n ( ) 1 as an approximation to e . We know from the previous chapter that the error is given by e p e n n n - = + + ( ) ( )! 1 1 1 1 ξ , where 0 1 < < ξ . Assume we know that e <3, and we have the estimate 0 1 3 1 - + e p n n ( ) ( )! . Meditate on this error estimate. It tells us that we can make this error as small as we like by choosing n sufficiently large. This is expressed formally by saying that the limit of p n ( ) 1 as n becomes infinite is e . This is the idea we shall study in this chapter. 10.2 Sequences A sequence of real numbers is simply a function from a subset of the nonnegative integers into the reals. If the domain is infinite, we say the sequence is an infinite sequence. (Guess what a finite sequence is.) We shall be concerned only with infinite sequences, and so the modifier will usually be omitted. We shall also almost always consider sequences in which the domain is either the entire set of nonnegative or positive integers. There are several notational conventions involved in writing and talking about sequences. If f Z : + R , it is customary to denote f n ( ) by f n , and the sequence itself by ( ) f n . (Here Z + denotes the positive integers.) Thus, for example, 1 n is the sequence
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f defined by f n n ( ) = 1 . The function values f n are called terms of the sequence. Frequently one sees a sequence described by writing something like 14 9 2 , , , , , K K n . This is simply another way of describing the sequence ( ). n 2 Let ( ) a n be a sequence and suppose there is a number L such that for any ε >0, there is an integer N such that | | a L n - < ε for all n > N . Then L is said to be a limit of the sequence, and ( ) a n is said to converge to L . This is usually written lim n n a L →∞ = . Now, what does this really mean? It says simply that as n gets big, the terms of the sequence get close to L . I hope it is clear that 0 is a limit of the sequence 1 n . From the discussion in the Introduction to this chapter, it should be reasonably clear that a limit of the sequence 1 1 2 1 6 1 + + + + K n ! is e . The graph of a sequence is pretty dreary compared with the graph of a function whose domain is an interval of reals, but nevertheless, a look at some pictures can help understand some of these definitions. Suppose the sequence ( ) a n converges to L . Look at the graph of ( ) a n : The fact that L is a limit of the sequence means that for any ε >0, there is an N so that to the right of N , all the spots are in the strip of width 2 ε centered at L . Exercises
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This note was uploaded on 10/27/2009 for the course MBA 1918272 taught by Professor Peter during the Fall '09 term at Aberystwyth University.

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cal10 - Chapter Ten Sequences, Series, and All That 10.1...

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