{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
10.2 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 .
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern