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 > R and 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 On completion you should be able to... have knowledge of infinite series and of the ratio test have knowledge of inequalities and of the factorial notation. explain what a power series is obtain the radius of convergence for a power series explain what a general power series is HELM (008): Workbook 6: Sequences and Series

2 . Power series A power series is simply a sum of terms each of which contains a variable raised to a non-negative integer power. To illustrate: x x + x 5 x x + x! + x! + are examples of power series. In the binomial series: + px + p(p ) x +! 6. we encountered an important example of a power series, 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 < R the power series is absolutely convergent and if x > R the power series is. At the two points x = R and x = R the power series may be convergent or. HELM (008): Section 6.4: Power Series

3 Key Point Convergence of Power Series For a power 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 R 0 convergent R x For any particular power series know, from the ratio test that + x p+ lim x p = lim + x p the value of R can be obtained using the ratio test. We x p is absolutely convergent if x < implying x < lim + and so R = lim +. Example (a) Find the radius of convergence of the series + x + x + x 4 + (b) Investigate what happens at the end-points x =, x = + of the region of absolute convergence. 4 HELM (008): Workbook 6: Sequences and Series

4 Solution (a) Here + x + x + x 4 + = x p p + so = p + In this case, R = lim p + p + = + = p + so the given series is absolutely convergent if x < and is if x >. (b) 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 <. Task Find the range of values of x for which the following power series converges: + x + x + x + First find the coefficient of x p : = = p Now find R, the radius of convergence: R = lim + = R = lim + = lim p+ p = lim () =. When x = ± the series is clearly. Hence the series is convergent only if < x <. HELM (008): Section 6.4: Power Series 5

5 . 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 the following properties: Key Point If P and P are power series with respective radii of convergence R and R then 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. Power series can also be differentiated and integrated on a term by term basis: Key Point If P is a power series with radius of convergence R then d dx (P ) and (P ) dx are each power series with radius of convergence R Example Using the known result that ( + x) p = + px + p(p ) x + x <,! choose p = and by differentiating obtain the power series expression for (+x). Solution ( + x) x = + + Differentiating both sides: ( ) x +! ( ) ( )! ( + x) = + Multiplying through by : ( + x) = x + x + ( ) x + ( ( ) ( ) x + ) ( ) x + This result can, of course, be obtained directly from the expansion for ( + x) p with p =. 6 HELM (008): Workbook 6: Sequences and Series

6 Task Using the known result that + x = x + x x + x <, (a) Find an expression for ln( + x) (b) Use the expression to obtain an approximation to ln(.) (a) Integrate both sides of dx + x = ( x + x ) dx = + x = x + x and so deduce an expression for ln( + x): dx = ln( + x) + c where c is a constant of integration, + x ( x + x ) dx = x x + x + k where k is a constant of integration. So we conclude ln( + x) + c = x x + x + k if x < Choosing x = 0 shows that c = k so they cancel from this equation. (b) Now choose x = 0. to approximate ln( + 0.) using terms up to cubic: ln(.) = 0. (0.) + (0.) ln(.) which is easily checked by calculator. HELM (008): Section 6.4: Power Series 7

7 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: R = lim + and the interval of convergence is now shifted to have centre at x = x 0 (see Figure 4 below). The series is absolutely convergent if x x 0 < R, diverges if x x 0 > R and may or may not converge if x x 0 = R. 0 x 0 R x 0 x 0 +R convergent Figure 4 x Task Find the radius of convergence of the general power 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 + = lim () p () p+ =. Hence R =, so the series is absolutely convergent if x <. 8 HELM (008): Workbook 6: Sequences and Series

8 Finally, decide on the convergence at x = (i.e. at x = and x = i.e. x = 0 and x = ): At x = 0 the 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 <. 0 convergent x. From the result Exercises x = + x + x + x +..., x < (a) Find an expression for ln( x) (b) Use this expression to obtain an approximation to ln(0.9) to 4 d.p.. Find the radius of convergence of the general power series (x+)+(x+) (x+) Find the range of values of x for which the power series + x 4 + x 4 + x converges. 4. By differentiating the series for ( + x) / find the power series for ( + x) / and state its radius of convergence. 5. (a) Find the radius of convergence of the series + x + x 4 + x s (b) Investigate what happens at the points x = and x = +. ln( x) = x x x x ln(0.9) (4 d.p.). R =. Series converges if < x <. If x = series diverges. If x = series diverges.. Series converges if 4 < x < ( + x) / = x + 5 x +... valid for x <. 5. (a) R =. (b) At x = + series diverges. At x = series converges. HELM (008): Section 6.4: Power Series 9