微积分教学资料——chapter81,82

1、Chapter 8Infinite Sequences and Series,8.1 Limits of Sequences of Numbers 8.2 Subsequences, Bounded Sequences, and Picards 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 Appli

2、cations of Power Series 8.9 Fourier Series 8.10 Fourier Cosine and Sine Series,A sequence can be thought of as a list of numbers written in a definite order:,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

3、 of the first few terms continues.,example,There are some sequences that dont 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:,It is obvious that the terms of the sequences n/(n+1) are approaching 1 as n becomes la

4、rge. 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.,For example:,A more precise version

5、of Definition 1 is as follows:,Geometrical interpretation:,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 Sa

6、ndwich Theorem for Sequences,holds for all beyond some index,And if then also.,snwid,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 and if is a function that is continuous at and defined at all,It is obvious that:,Geometrical meaning:,Example:,Solution:,Therefore,Since,Example:,For example:,8.2 Subsequences, Bounded Sequenc