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1、Chapter 1 Infinite SeriesGenerally, for the given sequence the expression formed by the sequence is called the infinite series of the constants term, denoted by , that is =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 1.1 Det
2、ermine whether the infinite series converges or diverges.While its possible to add two numbers, three numbers, a hundred numbers, or even a million numbers, its impossible to add an infinite number of numbers. To form an infinite series we begin with an infinite sequence of real numbers: , we can no
3、t form the sum of all the (there is an infinite number of the term), but we can form the partial sums.Definition 1.1.1If the sequence of partial sums has a finite limit L, We write and say that the series converges to L. we call L the sum of the series. If the limit of the sequence of partial sums d
4、ont exists, we say that the series diverges.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:(1) If then the converges, and (2)If then the diverges.Proof: the nth partial sum of the ge
5、ometric series takes the form Multiplication by x gives =Subtracting the second equation from the first, we find that . For this gives If then ,and this by equation . This proves (1).Now let us prove (2). For x=1, we use equation and device that Obviously, , diverges.For x=-1 we use equation and we
6、deduce If n is odd, then,If n is even, then The sequence of partial sum like this 0,-1,0,-1,0,-1.Because the limit of sequence of partial sum does not exist. By definition 1.1.1, we have the series diverges. (x=-1). For with we use equation . Since in this instance, we have. The limit of sequence of
7、 partial sum not exist, the series diverges. Remark the above series is called the geometric series. It arises in so many different contexts that it merits special attention. A geometric series is one of the few series where we can actually give an explicit formula for; a collapsing series is anothe
8、r. Ex.1.1.2 Determine whether or not the series converges Solution in order to determine whether or not this series converges we must examine the partial sum. Since We use partial fraction decomposition to write Since all but the first and last occur in pairs with opposite signs, the sum collapses t
9、o give Obviously, as this means that the series converges to 1. therefore EX.1.1.3 proves the following theorem:Theorem 1.1.1 the kth term of a convergent series tends to 0; namely if Converges, by definition we have the limit of the sequence of partial sums exists. Namely Obviously since, we have A
10、 change in notation gives. The next result is an obviously, but important, consequence of theorem1.1.1. Theorem 1.1.2 (A diverges test) if, or if does not exist, then the series diverges. Caution, theorem 1.1.1 does not say that if, and then converge. In fact, there are divergent series for which. F
11、or example, the series. Since it is sequence of partial sum is unbounded. So , therefore the series diverges. But EX.1.1.3 determine whether or not the series: Converges.Solution since, this series diverges. EX.1.1.4 Determine whether or not the series Solution 1 the given series is a geometric seri
12、es. , by proposition 1.1.1 we know that series converges. Solution 2 - (1-). By definition of converges of series, this series converges. EX.1.1.5 proofs the following theorem: Theorem 1.1.2 If the series converges, then (1) also converges, and is equal the sum of the two series.(2) If C is a real n
13、umber, then also converges. Moreover if then . Proof let Note that Since Then Theorem 1.1.4 (squeeze theorem)Suppose that both converge toand that for (k is a fixed integer), then also converges to .Ex.1.1.6 show that .Solution For since the result follows by the squeeze theorem.For sequence of vari
14、able sign, it is helpful to have the following result.EX1.1.7 prove that the following theorem holds.Theorem 1.1.5 If,Proof since from the theorem 1.1.4Namely the squeeze theorem, we know the result is true.Exercise 1.1(1) An expression of the form is called (2) A series is said to converge if the s
15、equence converges, where = 1. The geometric series converges if ; in this case the sum of the series is 2. If , we can be sure that the series 3. Evaluate.4. Evaluate.5. Show thatdiverges.Find the sums of the series 6-116. 7. 8. 9.10. 11.12. Derive the following results from the geometric series.Test the following series for convergence:13. 14.1.2 Series With Positive Terms1.2.1 The c
