微积分(下)英文教案资料.docx

k k 0 Chapter 1 Infinite Series Generally, for the given sequence a1, a2, a3 an, , the expression formed by the sequence a1 , a2, a3 an, , a1 a2 a3 an , is called the infinite series of the constants term, denoted by an , that is n1 an = a1 a2 a3 an , n1 Where the nth term is said to be the general term of the series, moreover, the nth partial sum of the series is given by Sn a1 a2 a3 an. 1.1 Determine whether the infinite series converges or diverges. While it ’ s possible to add two numbers, three numbers, a hundred numbers, or even a million numbers, it ’ s impossible to add an infinite number of numbers. To form an infinite series we begin with an infinite sequence of real numbers: a0, a1,a2,a3 , we can not form the sum of all the ak (there is an infinite number of the term), but we can form the partial sums 0 S0 a0 ak Sia0ak k 0S2aoa1a22akk 0S3a Si a0 ak k 0 S2 ao a1 a2 2 ak k 0 S3 ao ai a2 a3 3 ak k 0 Sn a0 a1 a2 a3 an n ak k 0 Definition 1.1.1 If the sequence { ⑴If Sn} of partial sums has a finite limit L, We write L ak k 0 and say that the series ak converges to L. we call L thek 0 sum of the series. If the limit of the sequence {Sn} of partial sums don t exists, we say that the series ak diverges. k 0 Remark it is important to note that the sum of a series is not a sum in the ordering sense. It is a limit. EX 1.1.1 prove the following proposition: Proposition1.1.1: Xkk 01 then the ak Xk k 0 k 0 ⑵If x 1, then the Xk diverges. Proof: the nth partial sum of the geometric series ak 0 takes the form Sn x1 x2 x3 xn 1 ① Multiplication by x gives 1 xSn x(1 x n 1、 1 x )= x Subtracting the second equation from the first, we find that (1 x)Sn xn. For x 1, this gives Sn If 1, then 0 ,and this by equation lnm Sn lnm This proves (1). Now let us prove ⑵. For x=1, we use equation and device that Sn n, Obviously, lim Sn n ak diverges. 0 For x=-1 we use equation ① and we deduce If n is odd, then Sn 0 , If n is even, then Sn 1. The sequence of partial sum Sn lik