Ch5.l:ReviewofPowerSeries* Finding the general solution of a linear differential equationdepends on determining a fundamental set of solutions of thehomogeneous equation.米So far, we have a systematic procedure for constructingfundamental solutions if equation has constant coefficients* For a larger class of equations with variable coefficients, wemust search for solutions beyond the familiar elementaryfunctions of calculus米The principal tool we need is the representation of a givenfunction by a power series.米 Then, similar to the undetermined coefficients method, weassume the solutions have power series representations, andthen determine the coefficients so as to satisfy the equation
Ch 5.1: Review of Power Series Finding the general solution of a linear differential equation depends on determining a fundamental set of solutions of the homogeneous equation. So far, we have a systematic procedure for constructing fundamental solutions if equation has constant coefficients. For a larger class of equations with variable coefficients, we must search for solutions beyond the familiar elementary functions of calculus. The principal tool we need is the representation of a given function by a power series. Then, similar to the undetermined coefficients method, we assume the solutions have power series representations, and then determine the coefficients so as to satisfy the equation
ConvergentPower Series A power series about the point xo has the formEa,(x-xo)and is said to converge at a point x ifmZa,(r-xo)limm-0n=lexists for that x* Note that the series converges for x = xo. It may converge forall x, or it may converge for some values of x and not others
Convergent Power Series A power series about the point x0 has the form and is said to converge at a point x if exists for that x. Note that the series converges for x = x0 . It may converge for all x, or it may converge for some values of x and not others. ( ) = − 1 0 n n n a x x ( ) = → − m n n n m a x x 1 0 lim
Absolute Convergence* A power series about the point xoEa,(x-xo)"is said to converge absolutely at a point x if the seriesa,(x-xo)-Ea, x- xoln=ln=lconverges.* If a series converges absolutely, then the series also converges.The converse, however, is not necessarily true
Absolute Convergence A power series about the point x0 is said to converge absolutely at a point x if the series converges. If a series converges absolutely, then the series also converges. The converse, however, is not necessarily true. ( ) = − 1 0 n n n a x x ( ) = = − = − 1 0 1 0 n n n n n n a x x a x x
Ratio Test* One of the most useful tests for the absolute convergence of apower seriesEa,(x-xo)"is the ratio test. If a, O, and if, for a fixed value of xan+(x-xo)n+1an=|x-xo/L,=x-x0llimanlima,(x-xo)"n-o0a.then the power series converges absolutely at that value of x if lx -xolL < 1 and diverges if lx - xolL > 1. The test is inconclusive ifIx - xoIL = 1
Ratio Test One of the most useful tests for the absolute convergence of a power series is the ratio test. If an 0, and if, for a fixed value of x, then the power series converges absolutely at that value of x if |x - x0 |L < 1 and diverges if |x - x0 |L > 1. The test is inconclusive if |x - x0 |L = 1. ( ) = − 1 0 n n n a x x lim , ( ) ( ) lim 0 1 0 0 1 1 0 x x L a a x x a x x a x x n n n n n n n n = − = − − − + → + + →
Radiusof Convergence* There is a nonnegative number p, called the radius ofconvergence, such that Za,(x - xo)n converges absolutely forall x satisfying Ix - xol < p and diverges for Ix - xol > p.长 For a series that converges only at xo, we define p to be zero米店For a series that converges for all x, we say that p is infiniteIf p > O, then x - xol < p is called the interval of convergence* The series may either converge or diverge when lx - xol = p.SeriesSeriesSeriesconvergesdivergesdivergesabsolutelyxo-pXo+pXoSeries-mayconvergeordiverge
Radius of Convergence There is a nonnegative number , called the radius of convergence, such that an (x - x0 ) n converges absolutely for all x satisfying |x - x0 | < and diverges for |x - x0 | > . For a series that converges only at x0 , we define to be zero. For a series that converges for all x, we say that is infinite. If > 0, then |x - x0 | < is called the interval of convergence. The series may either converge or diverge when |x - x0 | =