Ma 530 Introduction to Power Series

Transcription

1 Ma 530 Itroductio to Power Series Please ote that there is material o power series at Visual Calculus. Some of this material was used as part of the presetatio of the topics that follow. What is a Power Series? Recall that the geometric series ar coverges to a provided r. I this sectio we will r 0 take the ratio r to be a variable x. I particular, the geometric series 0 ax a ax ax ax 3 is a example of a power series. It coverges o the iterval x, which we say is cetered at 0 ad has radius. All this will be geeralized i this sectio. Defiitio A power series is a series of the form Sx c x a c 0 c x a c x a c 3 x a 3 0 where a,c 0,c,c, are costats. The umbers c are called the coefficiets of the series ad the umber a is called the ceter of the series. We also say that the series is expaded about x a ad that the th -order term is c x a. Remark I the 0 th -order term c 0 x a 0 c 0 we follow the covetio that x a 0, eve whe x a, i spite of the fact that 0 0 is ormally udefied. All the remaiig termsgoto0whex a, sicea a 0 0for 0. Therefore Sa c 0. Remark Very ofte we cosider power series cetered at x 0 which have the simpler form 0 c x c 0 c x c x c 3 x 3 For the series 3 0 x 5 3 6x 5 7 x 5 6x 5 3 idetify the ceter ad the coefficiet of the th -order term. Solutio: The ceter is at x 5. The coefficiet of the th -order term is c coefficiet of th -order term is 3 c So the

2 For the series. Idetify the ceter. x 0. Idetify the coefficiet of the 3 rd -order term Write out the terms of the series up to 3 rd -order... Radius of Covergece Recall that the geometric series ax a ax ax ax 3 0 coverges o the iterval x, which we say is cetered at 0 ad has radius. For a geeral power series, we have: Theorem (Power Series Covergece Theorem) The covergece of a power series Sx c x a c 0 c x a c x a c 3 x a 3 0 is characterized by oe of the followig three cases:. There is a positive umber R, called the radius of covergece, such that the series Sx coverges absolutely o the iterval a R,a R. The series may also coverge at oe, both or either of the edpoits a R ad a R.. The series Sx coverges oly for x a. I this case, we say that the radius of covergece is R The series Sx coverges for all real umbers x. I this case, we say that the radius of covergece is R. Note that a Power Series always coverges for x a, sice at x a 0 c x c 0. It may or may ot coverge for other values of x. Remark Notice that the ceter of the iterval a R,a R is at a, which is the ceter of the series. Further, the iterval a R,a R ca also be specified by either of the triple iequalities a R x a R or R x a R or by the absolute value iequality

3 x a R. Remark The radius of covergece is usually foud by applyig the ratio test to the series. Fid the ceter ad radius of covergece of the series x 5. 0 Solutio: The ceter is a 5. To fid the radius we apply the ratio test a a L lim a x 5 ad a lim x 5 x 5 lim x 5 x 5 x 5 x 5 lim x 5 The ratio test says the series coverges absolutely if L. I other words, or x 5, from which we idetify the radius of covergece as R. Thus, the series coverges absolutely o the iterval 3,7. Now you do it: Fid the ceter ad radius of covergece of the series 3 x The example ad exercise above illustrate the first case i the Power Series Covergece Theorem. Here are some examples of the other two cases ad of the use of the ratio test: Fid the ceter ad radius of covergece of the series x.! 0 Solutio: The ceter is a. To fid the radius we apply the ratio test: a! x ad a x! a L lim a lim lim! x lim! x!! x x 0 for all x. The ratio test says the series coverges absolutely if L. Sice L 0, the series coverges for all x ad the radius of covergece is R. 3

4 Fid the ceter ad radius of covergece of the series! x. 0 Solutio: The ceter is agai a. To fid the radius we apply the ratio test: a! x ad a a L lim a lim!!! x! lim x! x lim for all x except x 0 forx. x x The ratio test says the series diverges if L ad coverges if L. So, the series diverges for all x except x ad coverges for x. Thus, the radius of covergece is R 0. Ad some more exercises: Fid the ceter ad radius of covergece of the series 0... Fid the ceter ad radius of covergece of the series 0 Iterval of Covergece! x 3...! x..... Defiitio The Power Series Covergece Theorem implies that a power series Sx c x a c 0 c x a c x a c 3 x a 3 0 coverges o a iterval called its iterval of covergece. This iterval may cosist of a sigle poit a, the set of all real umbers,, or a fiite iterval which may be ope: a R,a R, closed:a R,a R, or half ope: a R,a R or a R,a R. We have see that the ceter a may be read off the series, ad the radius R may be determied usig the ratio test (or the root test). It remais to determie the covergece at the edpoits of the iterval of covergece. Remark It is much more importat to be able to determie the radius of covergece tha it is to be able to determie whether the series coverges at the edpoits of the iterval of covergece. Remark You caot use the ratio test or the root test to determie the covergece at the edpoits, because these tests fail whe L which is precisely at the edpoits of the

5 iterval of covergece. You must use some other test. The followig three examples illustrate the three cases of a ope, closed or half ope iterval of covergece. You try this oe first: Fid the iterval of covergece of the series 0 Fid the iterval of covergece of the series 0 Fid the iterval of covergece of the series 0 3 x x x For what values of x is the series x coverget? We shall use the Ratio Test agai. x x 6 3x 3 6. L lim x x lim x For the series to coverge we must have L, that is x Thus x or equivaletly x. or 3 x 5 or fially 3 x 5 Hece, the series coverges if 3 x 5 ad diverges for x 5 ad x 3. The cases whe x 3 ad x 5 must be tested separately. Whe x 3,the x, which diverges sice the th term of this series does ot go to zero as. Whe x 5,the x, which agai diverges. Why? 5

6 Thus this series coverges i 3 x 5 ad this is the Iterval of Covergece. For this example R. Fid the iterval of covergece ad the radius of covergece of 3 x Represetatio of a Fuctios as Power Series It is ofte useful to write a fuctio fx as a ifiite series. Fuctios Defied by Power Series Whe a power series coverges o a ope iterval a R,a R (of fiite or ifiite legth), It defies a fuctio fx c x a c 0 c x a c x a c 3 x a 3 0 o its iterval of covergece. The oly such fuctio we kow so far is the sum of the geometric series: x 0 x x x x 3 for x Operatios o Power Series Substitutio The easiest way to costruct a ew fuctio is to make a substitutio for the variable. I geeral, you ca replace the variable i a series by ay expressio which leaves it a power series. You must make the same substitutio i the iterval of covergece. Fid the power series expasio for cetered about x 0, ad fid its iterval of x covergece. Solutio: Ito the geometric series x x x x x 3 for x 0 substitute x t. Remember to also make the substitutio i the iterval of covergece: t t Next, replace t by x: 0 t t t 3 for t 6

7 x x 0 x x x 3 for x Fially, simplify the series ad the iterval of covergece: x x 0 x 6x 6x 6 for x This is the power series ad it coverges o the iterval,. Remark Oe ormally does the first two steps above at the same time by doig the replacemet x x ad ever metioig the variable t. However, if that is at all cofusig, do it i two steps. Fid the power series expasio for cetered about x, ad fid its 3 x iterval of covergece