4!üKö =kBhc SerieskBhcDefinition of SerieskBhc"A series is an infinite sum muS(kBhc°kBhc k = 0kBhc  asbuSkBhckkBhc‘ŠSeries are closely related to sequences, because one of the standard techniques used to understand series is to introduce the sequence of kBhc partial sumskBhc.kBhcssbuSkBhcnkBhc  = muS(kBhcnkBhc k = 0kBhc  asbuSkBhckkBhc,%In terms of the partial sum sequence, muS(kBhc°kBhc k = 0kBhc  asbuSkBhckkBhc  = tmiLkBhc nÏ°kBhc muS(kBhcnkBhc k = 0kBhc  asbuSkBhckkBhc  = tmiLkBhc nÏ°kBhc  ssbuSkBhcn kBhcÀ¹If the sequence of partial sums converges to a limit, we say that the original series converges to that limit. If the sequence of partial sums diverges, we say that the series diverges.kBhcãÜThis device for understanding series is especially useful in developing the theory of series, although it is of only limited practical usefulness. The problem with most series is that it is very difficult to compute what kBhcssbuSkBhcnkBhc  = muS(kBhcnkBhc k = 0kBhc  asbuSkBhck kBhcis as a function of n.kBhcThree important exampleskBhc¨¡There are a few isolated cases where it is possible to compute a partial sum by some means. In this section of the lecture we will encounter three such examples.kBhc?8Here is the most important example, the geometric series muS(kBhc°kBhc k = 0kBhc  ropxEkBhckkBhcb[The key to understanding the partial sums of the geometric series is the algebraic identitykBhcropxEkBhc n+1kBhc - 1 = GrapkBhc r - 1Grap†kBhc 1 + r + ropxEkBhc2kBhc  + ropxEkBhc3kBhc + É + ropxEkBhcnkBhc+$Using this identity we can show thatkBhcssbuSkBhcnkBhc  = muS(kBhcnkBhc k = 0kBhc  ropxEkBhckkBhc = 1 + r + ropxEkBhc2kBhc  + ropxEkBhc3kBhc + É + ropxEkBhcnkBhc  = carFQkBhcropxEkBhc n+1kBhc  - 1kBhc r - 1kBhc  kBhc=6This gives us an explicit formula for the partial sum kBhcssbuSkBhcnkBhc]V. With that formula we can compute the sum of the series as the limit of partial sums.  muS(kBhc°kBhc k = 0kBhc  ropxEkBhckkBhc  = tmiLkBhc nÏ°kBhc muS(kBhcnkBhc k = 0kBhc  ropxEkBhckkBhc  = tmiLkBhc nÏ°kBhc  ssbuSkBhcnkBhc  = tmiLkBhc nÏ°kBhc carFQkBhcropxEkBhc n+1kBhc  - 1kBhc r - 1kBhcProvided that VsbAkBhcrkBhc4- < 1, the latter limit exists and we get that muS(kBhc°kBhc k = 0kBhc  ropxEkBhckkBhc  = carF(kBhc1kBhc 1 - rkBhc Note that if kBhc r ³ 1kBhc" the original sum diverges.kBhcThe next example is muS(kBhc°kBhc k = 1carF@kBhc1kBhckGrapkBhc k + 1kBhce^This example can be handled by a trick: we use partial fractions to rewrite the partial sum askBhcssbuSkBhcnkBhc  = muS(kBhcnkBhc k = 1GrapjcarF$kBhc1kBhckkBhc  - carF(kBhc1kBhc k + 1kBhcohIf you write out the first few terms of a partial sum, you will see that almost all of the terms cancel. kBhcssbuSkBhc4kBhc  = GrapfcarF$kBhc1kBhc1kBhc  - carF$kBhc1kBhc2kBhc  + GrapfcarF$kBhc1kBhc2kBhc  - carF$kBhc1kBhc3kBhc  + GrapfcarF$kBhc1kBhc3kBhc  - carF$kBhc1kBhc4kBhc  + GrapfcarF$kBhc1kBhc4kBhc  - carF$kBhc1kBhc5kBhc = 1 - carF$kBhc1kBhc5kBhc,%In general, what happens here is thatkBhcssbuSkBhcnkBhc = 1 - carF(kBhc1kBhc n + 1kBhc Thus muS(kBhc°kBhc k = 1kBhc carF@kBhc1kBhckGrapkBhc k + 1kBhc  = tmiLkBhc nÏ°kBhc  ssbuSkBhcnkBhc  = tmiLkBhc nÏ°kBhc GrapDkBhc  1 - carF(kBhc1kBhc n + 1kBhc  = 1kBhc The final example is the kBhcharmonic series muS(kBhc°kBhc k = 1carF$kBhc1kBhckkBhc~wIt is possible to show that the harmonic series diverges by examining a carefully constructed sequence of partial sums.kBhcssbuSkBhc1kBhc  = 1kBhcssbuSkBhc2kBhc = 1 + carF$kBhc1kBhc2 kBhcssbuSkBhc4kBhc = 1 + carF$kBhc1kBhc2kBhc  + GrapfcarF$kBhc1kBhc3kBhc  + carF$kBhc1kBhc4kBhc > 1 + carF$kBhc1kBhc2kBhc  + GrapfcarF$kBhc1kBhc4kBhc  + carF$kBhc1kBhc4kBhc = 1 + carF$kBhc2kBhc2 kBhcssbuSkBhc8kBhc  = ssbuSkBhc4kBhc  + GrapÒcarF$kBhc1kBhc5kBhc  + carF$kBhc1kBhc6kBhc  + carF$kBhc1kBhc7kBhc  + carF$kBhc1kBhc8kBhc  > ssbuSkBhc4kBhc  + GrapÒcarF$kBhc1kBhc8kBhc  + carF$kBhc1kBhc8kBhc  + carF$kBhc1kBhc8kBhc  + carF$kBhc1kBhc8kBhc > 1 + carF$kBhc3kBhc2kBhc›”The same pattern continues through all the powers of 2. Every time we add more terms to the partial sum it continues to grow. Thus, in the limit as kBhcnkBhc gets very large kBhcssbuSkBhcnkBhc diverges. kBhc,%A necessary condition for convergence kBhc‘ŠIf a sequence of partial sums is going to have any chance to converge, the terms in the series have to get smaller as the summation index kBhckkBhc‰‚ gets larger. However, as the last two examples above demonstrate, simply having the terms of the series get smaller as the index kBhckkBhcVO grows is not strong enough to guarantee convergence. In the case of the series muS(kBhc°kBhc k = 1carF@kBhc1kBhckGrapkBhc k + 1kBhcthe individual terms carF@kBhc1kBhckGrapkBhc k + 1kBhc shrink in size as kBhckkBhcXQ gets large and the series converges. However, in the case of the harmonic series muS(kBhc°kBhc k = 1carF$kBhc1kBhckkBhcthe individual terms carF$kBhc1kBhckkBhc shrink as kBhckkBhc4- gets larger, but the series itself diverges.kBhc<5We can summarize this state of affairs by saying thattmiLkBhc kÏ°kBhc  asbuSkBhckkBhc  = 0kBhc is a kBhcnecessary conditionkBhc% for convergence of the series  muS(kBhc°kBhc k = 0kBhc  asbuSkBhck!kBhc but not a kBhcsufficient conditionkBhc for convergence. kBhcCombining series kBhc[TExcept for the complication caused by the fact that the upper limit in the summation muS(kBhc°kBhc k = 0kBhc  asbuSkBhckkBhc is kBhc°kBhcÿ, a series is basically a sum. All of the familiar algebraic rules that you know about sums like the distributive law continue to hold for series (as long as all of the series involved are convergent series). Here are some algebraic laws that we are goingkBhc to make use of. muS(kBhc°kBhc k = 0kBhc  c asbuSkBhckkBhc  = c muS(kBhc°kBhc k = 0kBhc  asbuSkBhck muS(kBhc°kBhc k = 0kBhc GrapSkBhcasbuSkBhckkBhc  + bsbuSkBhckkBhc  = muS(kBhc°kBhc k = 0kBhc  asbuSkBhckkBhc  + muS(kBhc°kBhc k = 0kBhc  bsbuSkBhck muS(kBhc°kBhc k = 0kBhc GrapSkBhcasbuSkBhckkBhc  - bsbuSkBhckkBhc  = muS(kBhc°kBhc k = 0kBhc  asbuSkBhckkBhc  - muS(kBhc°kBhc k = 0kBhc  bsbuSkBhck