16.4. Power Series. Introduction. Prerequisites. Learning Outcomes

Transcription

1 Power Series 6.4 Introduction In this section we consider power series. These are examples of infinite series where each term contains a variable, x, raised to a positive integer power. We use the ratio test to obtain the radius of convergence R, of the power series and state the important result that the series is absolutely convergent if x <R, if x >Rand may or may not be convergent if x = ±R. Finally, we extend the work to apply to general power series when the variable x is replaced by (x x 0 ). Prerequisites Before starting this Section you should... Learning Outcomes After completing this Section you should be able to... have knowledge of infinite series and of the ratio test have knowledge of inequalities and of the factorial notation. understand what a power series is obtain the radius of convergence for a power series know what a general power series is

2 . Power Series Apower series is simply a sum of terms each of which contains a variable raised to a positive integer power. To illustrate: x x + x 5 x 7 + +x + x! + x! + are examples of power series. Of course, in section 6., we encountered an important example of a power series, viz the binomial series: +px + p(p ) x +! p(p )(p ) x +! which, as we have already noted, represents the function ( + x) p as long as the variable x satisfies x <. A power series has the general form b 0 + b x + b x + = x p where b 0,b,b, are constants. Note that, in the summation notation, we have chosen to start the series at p =0. This is to ensure that the power series can include a constant term b 0 since x 0 =. The convergence, or otherwise, of a power series, clearly depends upon the value of x chosen. For example, the power series + x + x + x 4 + is convergent if x = (for then it is the alternating harmonic series) and if x =+ (for then it is the harmonic series).. The Radius of Convergence The most important statement one can make about a power series is that there exists a number, R, called the radius of convergence, such that if x <Rthe power series is absolutely convergent and if x >Rthe power series is. The relation x <Ris equivalent to R <x<r. At the two points x = R and x = R the power series may be convergent or. HELM (VERSION : March 8, 004): Workbook Level 6.4: Power Series

3 Key Point Convergence of Power Series For apower series x p with radius of convergence R then the series converges absolutely if x <R the series diverges if x >R the series may be convergent or at x = ±R See the following diagram. For any particular power series know, from the ratio test that + x p+ lim x p = lim + R 0 R convergent x p the value of R can be obtained using the ratio test. We x p is absolutely convergent if x < implying x < lim + x and so +. Example Find the radius of convergence of the series + x + x + x 4 + Solution Here + x + x + x + = x p 4 p + so = p + + = p + In this case, p + p + = so the given series is absolutely convergent if x < and is if x >. HELM (VERSION : March 8, 004): Workbook Level 6.4: Power Series

4 What happens at the end-points x =, x = +ofthe region of absolute convergence? At x =+the series is which is (the harmonic series). However, at x = the series is + + which is convergent (the alternating harmonic series). 4 Finally, therefore, the series + x + x + x 4 + is convergent if x<. Find those values of x for which the following power series converges: First find the coefficient of x p. = + x + x + x + Now find R, the radius of convergence + = + = lim = p p+ p = lim () =. When x = ± the series is clearly. Hence the series is convergent only if <x<.. Properties of Power Series Let P and P represent two power series with radii of convergence R and R respectively. We can combine P and P together by addition and multiplication. We find Key Point The sum (P +P ) and the product (P P ) are each power series with the radius of convergence being the smaller of R and R. HELM (VERSION : March 8, 004): Workbook Level 6.4: Power Series 4

5 Power series can also be differentiated and integrated on a term by term basis: d dx (P ) and Key Point (P )dx are each power series with radius of convergence R Example We have already seen that ( + x) p =+px + p(p ) x + x <! Choose p = and then, by differentiating, obtain the power series expression for ( + x). Solution ( + x) x =+ + differentiating both sides ( ) x +! ( + x) = + ( ) x + ( ( )( )! )( ) x + x + Multiplying through by : ( + x) = x + ( )( ) x + which can of course be obtained directly from the expansion for ( + x) p with p =. Using the known result that +x = x + x x + x < find an expression for ln(+x). Use this expression to obtain an approximation to ln(.) 5 HELM (VERSION : March 8, 004): Workbook Level 6.4: Power Series

6 Integrate both sides of +x = x + x dx +x = ( x + x )dx = so we conclude + k where k is a constant of integration ( x + x )dx = x x + x = ln( + x)+c where c is a constant of integration dx +x ln( + x)+c = x x + x + k if x < Choosing x =0shows that c = k so they cancel from this equation. Now choose x =0. tofind ln( + 0.) ln(.) =0. (0.) + (0.) ln(.) which is easily checked on your calculator. 4. General Power Series A general power series has the form b 0 + b (x x 0 )+b (x x 0 ) + = (x x 0 ) p Exactly the same considerations apply to this general power series as apply to the special series x p except that the variable x is replaced by (x x 0 ). The radius of convergence of the general series is obtained in the same way: and the interval of convergence is now shifted to have centre at x = x 0 (see diagram below). The series is absolutely convergent if x x 0 <R, diverges if x x 0 >Rand may or may not converge if x x 0 = R. + 0 x 0 R x 0 x 0 +R interval of convergence x HELM (VERSION : March 8, 004): Workbook Level 6.4: Power Series 6

7 Find the radius of convergence of the series (x ) + (x ) (x ) + First find an expression for the general term (x )+(x ) (x ) + = ( ) (x ) p () p so =() p Now obtain the radius of convergence lim = R = + + = lim () p () p+ =. Hence R =,sothe series is absolutely convergent if x < Finally, decide on the convergence at x =(i.e. at x = and x =i.e. x =0 and x = ). At x =0the series is +++ which diverges and at x =the series is + which also diverges. Thus the given series only converges if x < i.e. 0 <x<. lim 0 interval of convergence x 7 HELM (VERSION : March 8, 004): Workbook Level 6.4: Power Series