1、Concepts and Properties of Series with Constant Terms12Infinite SeriesIf the terms of the sequence are numbers then the series is called a series with constant terms(or simply a series级数级数).Suppose we have an infinite sequence 12naaaLLor1nna is called an infinite series or simply a series,and an is
2、called the general term通项通项 of the series.It is convenient to use sigma notation to write the series as1,nna 1,kka orna A useful shorthandwhen summation from 1 to is understood12,na aaLLAn express of the form3Infinite SeriesWe begin by asking how to assign meaning to an expression like111124816LThe
3、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 look for a pattern in how these“partial sum部分和部分和”grow.4Partial SumThe sum of the first n terms of the series121,(1,2,)nnnkkSaaaanLLis called the partial sum部分和部分和 of th
4、e series.The partial sum of the series form a sequence112121,nnkkSaSaaSaLLof real numbers,each defined as a finite sum.5Convergence and DivergenceThe examples of a bouncing ball is converge.The deference between the sum and the partial sum of the series,1nnkk nRSSa is called the remainder 余项余项 of th
5、e series.Definition(Convergence and Divergence of a Series)The series1kka converges if the sequence of partial sumconverges In this case,the limit of the sequence,is called the sum of the series,denoted by nS1limlim,nnknnkSaS1.kkaS Otherwise,we say the series diverges.Harmonic series6A series of the
6、 form11111123 nnnLLis called a harmonic series调和级数调和级数.Since the partial sum of the series1111,1,2,23 nSnnLLdiverges,the harmonic series is divergent.7Geometric Series 几何级数几何级数A series of the form210(0)nnnaaqaqaqaqaLLis called a geometric series(or series of equal ratios).It is easy to see that the
7、partial sum of the series is21(1),1,1,1.nnnaqqSaaqaqaqqnaq LHence,when|1,q (1)limlim11nnnnaqaSqq Thus the series converges.Otherwise,the series diverges(Why?).8Repeating Decimals Express the repeating decimal 5.23 23 23 as the ratio of two integers.Solution 232323235.232323.5100(100)(100)L2231151100
8、1001002315100 0.99L235185.99999NoteUnfortunately,formulas like the one for the sum of a convergent geometric series are rare,and we usually have to settle for an estimate of a series sum(more about this later).The next example,however,is another case in which we can find the sum exactly.10Example:A
9、Non-geometric but Telescoping SeriesSolutionWe look for a pattern in the sequence of partial sums thatmight lead to a formula for sk.The key is partial fractions.Since 111,(1)1k kkkwe have11111(1)1 22 3(1)knn nkk Lwhich can be written as 11111112231kSkk LFind the sum of the series11.(1)nn n(the part
10、ial sum)11Solution(continued)Removing parentheses and canceling the terms of positive sign collapses the sum to 11.1kSk We now see that1ks as.k The seriesconverges,and its sum is 1.Thus111.(1)nn n Example:A Non-geometric but Telescoping SeriesFind the sum of the series11.(1)nn n12(a)The series221149
11、nnn LLdiverges because the partial sums grow beyond every number L.After n=1,the partial sum sn=1+4+9+n2 is greater than n2.(b)The series112341123nnnnn LLdiverges because the partial sums eventually outgrow very pre-assigned number.Each term is greater than 1,so the sum of n terms is greater than n.
12、Example:Divergent series13Properties of SeriesIf naAand nbBare convergent series,then1.Sum Rule:nnnnababAB2.Difference Rule:nnnnababAB3.Constant Multiple Rule:nnkakakA(any number k)4.Order Rule:If(),nnabnNthen11.nnnnab Properties of Series14Theorem (Associative Property)If a series converges,then it
13、s sum is not changed when we add arbitrarily some brackets among the terms of the series(provided that the order of the terms is maintained).Proof (Page 295)15Properties of SeriesIf and1nna converges,thennn ka converges for any k 1,1211.nknnn kaaaaa LConversely,if nn ka converges for any k 1,then 1n
14、na converges.Theorem (Adding or Deleting Terms)Deleting,adding or changing any finite number of terms of a series does not change the convergence or divergence of the series.16Theorem (Limit of the nth Term of a Convergent Series)If 1nna converges,then 0.na Properties of SeriesObserve that limnnamus
15、t equal zero if the series1nna converges.To see why,let S represent the series sum and 12nnSaaaLthe nth partial sum.When n is large,both sn and sn-1 are close to S,so their difference,an,is close to zero.More formally,10nnnaSSSS Difference Rule for sequenceThis theorem can be used to test if a serie
16、s is divergent and we name it as nth-Term Test for Divergence.A necessary condition for convergence of a series17Properties of Series(a)21nn diverges because 2.n(b)11nnn diverges because11.nn(c)11(1)nn diverges because1lim(1)nn does not exist.(d)125nnn diverges because1lim0.252nnn 18Properties of Se
17、ries0,na but the Series DivergesThe series2 terms4 terms2 terms11111111112244442221 1 11nnnn LLL14444 4244444 3144444424444443LLdiverges even though its terms form a sequence that converges to 0.19Cauchys convergence principleThe necessary and sufficient condition for a series to be convergent is th
18、at0,NN ,such that N,p inequality 12|nnnpaaa Lholds for all n N.This conclusion can be obtained directly by applying the Cauchy convergence principle for sequences to the sequence of partial sum of the series.This theorem is a very powerful tool,which is used to justify the convergence of a series.20
19、Cauchys convergence criteria Prove that the series211nn converges.Proof,N,n p we have222111(1)(2)()nnnpL111(1)(1)(2)(1)()n nnnnpnpL1111111121nnnnnpnpL111nnpn Hence,for any given positive,take1N and then the inequality21Cauchys convergence criteriaProof(continued)222111(1)(2)()nnnp Lholds for all.nN Then,by the Cauchys convergence principle,the series converges.Prove that the series211nn converges.
