微积分英文版教学课件：8-1 Limits of Sequences of Numbers

1、Chapter 8Infinite Sequences and Series8.1 Limits of Sequences of Numbers8.2 Subsequences, Bounded Sequences, and Picards Method8.3 Infinite Series8.4 Series of Nonnegative Terms8.5 Alternating Series,Absolute and Conditional Convergence8.6 Power Series8.7 Taylor and Maclaurin Series8.8 Applications

2、of Power Series8.9 Fourier Series8.10 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 NumbersFor example:Find a formula for the general term ofthe sequence assuming that the patternof the first fe

3、w terms continues.exampleThere are some sequences that dont have a simple defining equation.The sequence where is the population of the world as of January1 in the year n. exampleFor example:It is obvious that the terms of the sequences n/(n+1) are approaching 1 as n becomes large. In fact the diffe

4、rence 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 of Definition 1 is as fo

5、llows:Geometrical interpretation:ExampleProve that Proof1.Guessing a value for Let be a given positive number.We should chooseIfWe haveSo when2.Showing that this works.givenLetIf then Therefore , by the definition of a limit,Theorem:Example:Solution:Find The Sandwich Theorem for Sequencesholds for a

6、ll beyond some index And if then also. snwid Let and be sequences of real numbers. IfSolution:that is: then:Example:Then:The Continuous Function Theorem for Sequencesthen . 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:ThereforeSinceExample:For example:8.2 Subsequences, Bounded Sequ