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1、1Lecture 11Series of Constant TermsA Bouncing BallDrop a ball from H meters above a flat surface. Each time the ballhits the surface after falling a distance h, it rebounds a distance rh, where r is positive but less than 1. find the total distance the ball travels up and down.It is easy to see that
2、 the total distance is2322 2s H Hr Hr Hr= + + +The question is that how to calculate the2sum with infinite terms.It is well known that, if there has finite terms2322 2 22(1 )1nnns H Hr Hr Hr HrHr rHr=+ + + +=+It is clear that2(1lim)2lim .11nnnnHr rHrsH Hrr=+ =+ Repeating DecimalsExpress the repeatin
3、g decimal 5.23 23 23 as the ratio of two integers.Solution 2323 23 235.232323. 5100 (100) (100)=+223 1 1351100 100 100=+ + + +23 1 23 51855.100 0.99 99 99=+ =+ =Infinite SeriesIf the terms of the sequence are numbers then the series is called a The following representation12 naa a+ +or1nna=which con
4、sists of all the terms of a sequence an and connected successively by the plus sign, is called an infinite series or simply aseries, and anis called the general term of the series. Definition4series of constant terms. If there is little chance of confusion, it may be called simply a series.It is con
5、venient to use sigma notation to write the series as1,nna=1,kka=ornaA useful shorthandwhen summation from 1 to is understoodInfinite SeriesA finite sum of real numbers always produces a real number, but an infinite sum of real numbers is something else entirely. This is why we need a careful definit
6、ion of infinite series.We begin by asking how to assign meaning to an expression like111 11+ +524816The way to do so is not to try to add all the terms at once (we cannot)but rather to add the terms one at a time from the beginning and lookfor a pattern in how these “partial sum” grow.Infinite Serie
7、sFirst: 1 1s =21Second: 2112s = +122Partial Sum Value1111 1124816nna= + +6Third:311124s = +124 nth111 1124 2n ns=+ +1122n2Partial SumDefinition The sum of the first n terms of the series121,( 1,2,)nnnkkSaa a a n=+= =is called the partial sum of the series.7The partial sum of the series form a sequen
8、ce11 2121,nnkksa saa s a=+of real numbers, each defined as a finite sum.Convergence and DivergenceDefinition (Convergence and Divergence of a series)If the sequence of partial sums of a series1kka= has a limit S asn approaching infinite, we say that the series converges to the sum S,and we write1231
9、.nkkaaa a a S+ += =8=Otherwise, we say that the series diverges.The deference between the sum and the partial sum of the series,1nn kknR SS a=+= = is called the remainder of the series.Geometric SeriesA series of the form210(0)nnnaaqaq aq aq a=+ + += is called a geometric series (or series of equal
10、ratios).It is easy to see that the partial sum of the series is21(1 ),1,1nnnaqqS a aq aq aq q =+ + + + = 9,1.na q =Hence, when |1,q 1, and12 11.nknkaaa a a =+ + 16nn= =Conversely, if nnka= converges for any k 1, then 1 nna=converges.Theorem (Adding or Deleting Terms)Adding, deleting or changing any
11、finite number of terms of a seriesdoes not change the convergence or divergence of the series.Properties of SeriesWhenever we have two convergent series, we can add them term byterm, subtract them term by term, or multiply them by constants to makea new convergent series.Theorem (Properties of Combi
12、ning Series)If naA=andnbB= are convergent series, then1 SRl ( )bbAB+17. Sum Ru e:nn n na b a bA+ = + =2. Difference Rule: ()nn n nab a b AB =3. Constant Multiple Rule: nnka k a kA= (any number k)4. Order Rule: If (),nnabnN+ then11.nnnnab=Properties of SeriesTheorem (Associative Property)If a series
13、converges, then its sum is not changed when we add arbitrarily some brackets among the terms of the series so long as we do not change the order of the terms.1812 345()( )+ +aa aaaThen lim lim .mnmnss = =,21s=,52s=,93s= ,nms=4Cauchys convergence principleTheorem (Cauchys convergence principle)The ne
14、cessary and sufficient condition for a series to be convergentis that 0, NN+ , such that Np+ , inequality 12|nn npaa a + + N.This theorem is a very powerful tool, which is used to justify the convergence of a series.Cauchys convergence principleExample Prove that the series211nn= converges. Proof ,N
15、,np+ we have22 211 1(1)(2) ( )nn np+ +11 120(1)(1)(2) ( 1)( )nn n n np npThen, by the Cauchys convergence principle, the series converges.Series of Non-negative TermsGiven a series na , we have two questions.1. Does the series converge?2. If it converges, what is the sum?From now on, we study series that do not have negative terms. This kind of series is also called as series of positive terms.The reason for this restriction is that the partial sums of these series22Th reas this restriction th su seriesform non-decreasing seque
