Theorems About Power Series

Transcription

1 Physics 6A Witer 20 Theorems About Power Series Cosider a power series, f(x) = a x, () where the a are real coefficiets ad x is a real variable. There exists a real o-egative umber R, called the radius of covergece such that. If R = 0, the the series i eq. () coverges for x = 0 ad diverges for ay o-zero real value of x. 2. If R =, the the series i eq. () coverges absolutely for ay (fiite) real umber x. 3. If 0 < R <, the the series i eq. () coverges absolutely for every real umber x such that x < R, ad diverges for every real umber x such that x > R. I may cases, R ca be determied by the ratio test, which yields R = lim a + a Examples of the three possible cases exhibited above are:. (2) (i)! x, (ii) x!, (iii) x. I particular, usig eq. (2), it follows that for the three series listed above, R = 0 for series (i), R = for series (ii) ad R = for series (iii). The iterval of covergece for the series i eq. () is defied to be the set of all possible values of x for which the series coverges. Note that if if 0 < R <, the the covergece properties of eq. () for x = R ad x = R are ot specified, If the limit i eq. (2) does ot exist, the a differet test, called the root test, ca be used to determie the radius of covergece. The root test yields /R = lim a /. If this limit fails to exist, oe ca modify the test slightly by employig the subsequece obtaied from the {a } for which the root test yields the largest possible value i the limit of. If both the ratio test ad the root test apply, oe ca show that they both yield the same value for the radius of covergece R.

2 ad must be determied by other meas. Thus, the iterval of covergece may or may ot iclude oe or both of the edpoits of the iterval R x R. The possible covergece properties at a edpoit are: absolute covergece, coditioal covergece or divergece. Theorem : The power series f(x) = a x is absolutely coverget for x < R, where R is the radius of covergece. Moreover, a x is cotiuous ad ifiitely differetiable withi the iterval of covergece, x < R. Proof: The covergece properties of the power series are a cosequece of the ratio test. The proof of cotiuity ad differetiability ca be foud i the refereces at the ed of this ote. Theorem 2: If the power series f(x) = a x is coverget at x = R, the it is a cotiuous fuctio withi the iterval of covergece icludig the edpoit at x = R. I this case, we have f(r) = lim x R a x = lim a x = x R a R, where lim x R meas that x approaches R from the left, i.e. from iside the iterval of covergece, x < R. That is, i this case it is permissible to iterchage the order of the limit ad the ifiite sum. Likewise, if the power series is coverget at x = R, the it is a cotiuous fuctio withi the iterval of covergece icludig the edpoit at x = R. I this case, we have f( R) = lim x R + a x = lim a x = ( ) a R, x R + where lim x R + meas that x approaches R from the right, i.e. from iside the iterval of covergece, x < R. Theorem 2 is kow as Abel s theorem. As a example of its applicatio, cosider the power series, l( + x) = ( ) +x, x <. (3) I this case, the radius of covergece is R =. Moreover, if we set x = above, the resultig series is coditioally coverget (as a cosequece of the alteratig series test). Thus, the power series for l( + x) is cotiuous at x =, which allows us to coclude that: l 2 = ( ) +. 2

3 The coverse of Abel s theorem is sometimes false. As a example, we cosider the ifiite geometric series, + x = ( ) x. (4) Settig x = above yields the diverget series Hece, the coditios of Abel s theorem are ot satisfied, i which case we caot coclude that ( ) x is cotiuous at x =. I particular, for x =, the left had side of eq. (4) yields. Although oe ca make a case for assigig to the series , the latter series is clearly ot coverget accordig to the stadard mathematical defiitio of covergece. Theorem 3: Cosider a power series f(x) = a x with radius of covergece R. The, term-by-term differetiatio ad itegratio of the power series is permitted, ad does ot chage the radius of covergece. That is, df = d a x d = a x = f(x) = a x = a x = a x, x < R, (5) a x + +. x < R. (6) I particular, for values of x withi the iterval of covergece, x < R, it is permissible to iterchage the order of the ifiite summatio ad the differetiatio or itegratio. This feature is oe of the reasos that power series are so ice they behave for the most part like ordiary polyomials. Proof: This theorem is a simple cosequece of the ratio test. Note that the ratio test is icoclusive at the edpoits of the iterval of covergece, so that the covergece properties at x = R ad x = R must be separately ivestigated. Although a power series, its derivative ad its itegral possess the same radius of covergece, this does ot mea that they have the same iterval of covergece. I particular, the itervals of covergece of the power series represetatios of f(x), df/ ad f(x) ca differ at the edpoits of the iterval of covergece. I geeral, by differetiatig a fuctio defied by a power series with radius of covergece R, we may lose covergece at a edpoit of the iterval of covergece of f(x). I cotrast, by itegratig a fuctio defied by a power series with radius of covergece R, we may gai covergece at a edpoit of the iterval of covergece of f(x). O the other had, the series compariso I geeral, if f(x) = f (x) is a poitwise coverget sum, it may happe that the itegral of the ifiite sum is ot equal to the ifiite sum of the itegrals, ad/or the derivative of the ifiite sum is ot equal to the ifiite sum of the derivatives. However, this caot happe for a power series whe x lies withi the iterval of covergece. 3

4 test implies if f(x) diverges at a edpoit, the df/ must also diverge at that edpoit, whereas if f(x) coverges at a edpoit, the f(x) must also coverge at that edpoit. The followig two examples are istructive. First, we defie the dilogarithm Li 2 (x) via the power series, Li 2 (x) x, x. (7) 2 The ratio test implies that the radius of covergece is R =, ad the p-series test implies that the power series coverges absolutely at both edpoits of the iterval of covergece. Takig a derivative of eq. (7) yields d Li 2(x) = d x = d 2 2 x = x, x <. (8) At x = the resultig series is the alteratig harmoic series which coverges, whereas at x = the resultig series is the harmoic series which diverges. Usig Abel s theorem, we ca exted the domai of validity of eq. (8) to iclude the edpoit x = (but ot the edpoit x = ). That is, eve though the series give by eq. (7) is coverget at x =, the series represetatio of the derivative of Li 2 (x) is diverget at x =. Usig eqs. (3) ad (8), it follows that: d Li l( x) 2(x) =. (9) x Strictly speakig, this result is oly valid i the rage x <. For our secod example, we start with the ifiite geometric series give i eq. (4), which diverges at both edpoits of the iterval of covergece. Computig the itegral of eq. (4) yields: = l( + x) = + x = ( ) x+ ( ) x = + = x ( ) ( ) x, x <. (0) At x = + the resultig series is the alteratig harmoic series which coverges, whereas at x = the resultig series is the egative of the harmoic series which diverges. Usig Abel s theorem, we ca exted the domai of validity of eq. (0) to iclude the edpoit x = (but ot the edpoit x = ). That is, eve though the ifiite geometric series give i eq. (4) is diverget at x =, the series represetatio of the itegral of /( + x) is coverget at x =. 4

5 Theorem 4: Give two power series with radii of covergece R ad R 2, respectively, i.e. f (x) = a x, x < R, () f 2 (x) = b x, x < R 2, (2) the the sum ad product of the two power series are give respectively by: f (x) + f 2 (x) = (a + b )x, x < R, (3) f (x)f 2 (x) = k=0 a k b k x, x < R, (4) where the radius of covergece of the sum ad of the product is at least as large as the miimum of R ad R 2, i.e. R mi{r, R 2 }. The subtractio of two series is the defied simply by chagig the sigs of all the b above before addig the two series. The divisio of the two series, f (x)/f 2 (x), ca be performed if ad oly if b 0 0. Assumig that this coditio holds, f (x) f 2 (x) = c x, x < R, (5) where the radius of covergece satisfies R mi{r, R 2, x 0 }, with x 0 idetified as the zero of f 2 (x) earest to x = 0. The coefficiets c i eq. (5) are determied recursively usig: [ c 0 = a 0 b 0, c = b 0 a ] b k c k k= for =, 2, 3,.... I the geeric case, R = mi{r, R 2 } ad R = mi{r, R 2, x 0 }. However, i special cases the radius of covergece may be larger. Here is oe such example: z = z, z <, (6) z (2 z)( z) = z z = ( ) z 2, z <, (7) 2 have radii of covergece R = R 2 =. Nevertheless, the sum of the two series defied i eqs. (6) ad (7) has a radius of covergece R = 2 > mi{r, R 2 }, z = ( z ), z <

6 Theorem 5: The power series represetatio of a fuctio, f(x) = a x, with a o-zero radius of covergece x < R, is uique. Proof: This is a cosequece of Taylor s theorem i calculus, which provides a explicit formula for the coefficiets of a power series, a = d f.! x=0 Refereces All of the results obtaied i these otes ca be foud i stadard mathematical refereces. I particular, the followig two refereces are elemetary ad highly readable:. Earl D. Raiville, Ifiite Series (The Macmilla Compay, New York, 967). 2. O.E. Staaitis, A Itroductio to Sequeces, Series, ad Improper Itegrals (Holde-Day, Ic., Sa Fracisco, 967). At a slightly higher level, but still accessible, I also recommed: 3. T.J.I a. Bromwich, A Itroductio to the Theory of Ifiite Series (Macmilla & Co. Ltd., Lodo, 959). 4. Korad Kapp, Theory ad Applicatio of Ifiite Series (Dover Publicatios, Ic., Mieola, NY, 990). 5. Bria S. Thomso, Judith B. Brucker ad Adrew M. Brucker, Elemetary Real Aalysis (Pretice-Hall, Ic., Eglewood Cliffs, NJ, 200). 6