# 62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +

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1 62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of the series I geeral, the series will coverge or diverge, depedig o the choice of x. The power series always coverges for x = 0 to the umber c 0. Example 98. For what values of x does the power series series x / coverge? Solutio: By the root test, x = x x as

2 62. POWER SERIES 125 So, the series coverges for all 1 < x < 1 ad diverges as x > 1 or x < 1. The root test is icoclusive for x = ±1. These values have to ivestigated by differet meas. For x = 1, the power series becomes the harmoic series 1/ which is diverget. For x = 1, the power series becomes the alteratig harmoic series ( 1) / which is coverget. Thus, the power series coverges if x [ 1, 1) ad diverges otherwise. Give a umber a, cosider a power series i the variable y = x a c y = c (x a) It is also called a power series cetered at a or a power series about a. Let S be the set of all values of x for which a power series i x coverges ad let S a be the set of all values of x for which the correspodig power series i (x a) coverges. What is the relatio betwee S ad S a? Sice the series are obtaied from oe aother by merely shiftig the value of the variable by a umber a, x x a, the set S a is therefore obtaied by addig the umber a to every elemet of S: (60) x S a x a S = S a = {x x a S} For example, the series (x 2) / coverges if x 2 [ 1, 1) or x [1, 3) ad diverges otherwise by Example 98. Thus, the problem of fidig the set S a is equivalet to the problem of fidig the set S Power series as a fuctio. Suppose that a power series i x coverges o a set S. The it defies a fuctio o S: f(x) = c x, x S The set S is called the domai of such a fuctio. Fuctios defied by power series are most commo i applicatios. May of them have special otatios (like elemetary fuctios si, cos, exp, etc). Their properties are well studied. I what follows it will be show that familiar elemetary fuctios si(x), cos(x), exp(x), etc ca also be represeted as power series. Example 99. Fid the domai of the Bessel fuctio of order 0 that is defied by the power series J 0 (x) = ( 1) 2 2 (!) 2 x2

3 SEQUENCES AND SERIES where, by commo covetio, 0! = 1. Solutio: Sice a = c x 2 cotais the factorial, the ratio test is more coveiet: a +1 a = x 2 c +1 c = x (!) 2 2 2(+1) (( + 1)!) 2 = x ( + 1) 2 0 as. So, the series coverges for all x. Values of a fuctio defied by a power series ca be estimated by partial sums which are polyomials i the variable x: f(x) f (x) = c k x k = c 0 + c 1 x + c 2 x c x k=0 Thus, partial sums defie a sequece of polyomials that coverges to the fuctio o S, f (x) f(x) for all x S. The accuracy of the approximatio is determied by the remaider R (x) = f(x) f (x). The accuracy assessmet is discussed i Sectio Sice the remider R (x) is a fuctio o S, the error of the approximatio is ot geerally uiform, i.e. it depeds o x Radius of covergece. The set S o which a power series is coverget is a importat characteristic ad its properties have to be studied. Lemma 2. (Properties of a power series) (i). If a power series c x coverges whe x = b 0, the it coverges wheever x b. (ii). If a power series c x diverges whe x = d 0, the it diverges wheever x d. Proof: If c b coverges, the by the ecessary coditio for covergece, c b 0 as. This meas, i particular, that for ε = 1 there exists a iteger N such that c b < ε = 1 for all > N. Thus, for > N, c x = c b x = c b x < x b b b which shows that the series c x coverges by compariso with the geometric series q where q = x/b ad x/b < 1 or x < b. Suppose that c d diverges. If x is ay umber such that x > d, the c x caot coverge because, by part (i) of the lemma, the covergece of c x implies the covergece of c d. Therefore c x diverges. This lemma allows us to establish the followig descriptio of the set S.

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