微积分教学课件chapter8.1,8.2.ppt

Chapter 8Infinite Sequences and Series A sequence can be thought of as a list of numbers written in a definite order: For example: It is obvious that the terms of the sequences {n/(n+1)} are approaching 1 as n becomes large. In fact the difference can be made as small as we like by taking n sufficiently large. We indicate this by writing In general ,we have the definition: The following figure illustrates Definition 1 by showing the graphs of two sequences that have the limit L. A more precise version of Definition 1 is as follows: Geometrical interpretation: It is obvious that: Example: * * 8.1 Limits of Sequences of Numbers 8.2 Subsequences, Bounded Sequences, and Picard’s Method 8.3 Infinite Series 8.4 Series of Nonnegative Terms 8.5 Alternating Series,Absolute and Conditional Convergence 8.6 Power Series 8.7 Taylor and Maclaurin Series 8.8 Applications of Power Series 8.9 Fourier Series 8.10 Fourier Cosine and Sine Series is the nth term. 8.1 Limits of Sequences of Numbers For example: Find a formula for the general term of the sequence assuming that the pattern of the first few terms continues. example There are some sequences that don’t have a simple defining equation. The sequence where is the population of the world as of January 1 in the year n. example For example: Example Prove that Proof 1.Guessing a value for Let be a given positive number. We should choose If We have So when 2.Showing that this works. given Let If then Therefore , by the definition of a limit, Theorem: Example: Solution: Find The Sandwich Theorem for Sequences holds for all beyond some index And if then also. s?nwid?] Let and be sequences of real numbers. If Solution: that is: then: Example: Then: The Continuous Function Theorem for Sequences then . Let be a sequences of real numbers. If