Let {ak}be a sequence of numbers.The series of the form ∑a(x-a) k=0 k=0 are called the power series in x and (x-a)respectively Example:The Taylor series x is a power series k! Sals,Hille,Etgen Calculus One and Several Varibles Main Menu Copy right 2007 John Wiley Sors,Inc.All rights reserved
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Let {ak} be a sequence of numbers. The series of the form are called the power series in x and (x-a) respectively. , 0 k= k ak x = − 0 ( ) k k ak x a Example: The Taylor series is a power series. =0 ( ) ! (0) k k k x k f
Power Series DEFINITION 12.8.1 A power series akx is said to converge (①at cif∑akck converges;: (il)on the set S if∑akx converges at each x∈S. Salas,Hille,Etgen Calculus:One and Several Varisbles Main Menu Copy right 2007 John Wiley Sors,Inc.All rights reserved
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Power Series
THEOREM 12.8.2 If ax converges at c0,it converges absolutely at all x with cl. Ifax diverges at d,then it diverges at allx with>dl. Salas,Hille,Etgen Calculus:One and Several Varisbles Main Menu Copy right 2007 John Wiley Sors,Inc.All rights reserved
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved
Power Series There are exactly three possibilities for a power series: Case 1.The series converges only at x =0.This is what happens with ∑kx Forx0,kkxk0,and so the series cannot converge. Case 2.The series converges absolutely at all real numbers x.This is what happens with the exponential series Case 3.There exists a positive number r such that the series converges absolutely forr.This is what happens with the geometric series ∑ Here there is absolute convergence for1. Salas,Hille,Etgen Calculus:One and Several Varisbles Main Menu Copy right 2007 John Wiley Sors,Inc.All rights reserved
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Power Series There are exactly three possibilities for a power series: Case 1. The series converges only at x = 0. This is what happens with For x ≠ 0, kk x k 0, and so the series cannot converge. Case 2. The series converges absolutely at all real numbers x. This is what happens with the exponential series Case 3. There exists a positive number r such that the series converges absolutely for |x| r . This is what happens with the geometric series → . k k k x. ! k x k Here there is absolute convergence for |x| 1. . k x
Power Series Associated with each case is a radius of convergence: In Case 1,we say that the radius of convergence is 0. convergence only at O 0 case 1:radius of convergence =O In Case 2,we say that the radius of convergence is oo. convergence everywhere 0 case 2:radius of convergence =oo In Case 3,we say that the radius of convergence is r. divergence convergence divergence 0 case 3:radius of convergence Figure 12.8.1
Main Menu Power Series Associated with each case is a radius of convergence: In Case 1, we say that the radius of convergence is 0. In Case 2, we say that the radius of convergence is ∞. In Case 3, we say that the radius of convergence is r
Example: Σ Salas,Hille,Etgen Calculus:One and Several Varisbles Main Menu Copy right 2007 John Wiley Sors,Inc.All rights reserved
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Example: − k k x k ( 1)
Example: 2点 Salas,Hille,Etgen Calculus:One and Several Varisbles Main Menu Copy right 2007 John Wiley Sors,Inc.All rights reserved
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Example: k x k 2 1
Example: 2。 k Salas,Hille,Etgen Calculus:One and Several Varisbles Main Menu Copy right 2007 John Wiley Sors,Inc.All rights reserved
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Example: k k x k 6
Example: Salas,Hille,Etgen Calculus:One and Several Varisbles Main Menu Copy right 2007 John Wiley Sors,Inc.All rights reserved
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Example: k x k k (3 )! !
Example: -2y Salas,Hille,Etgen Calculus:One and Several Varisbles Main Menu Copy right 2007 John Wiley Sors,Inc.All rights reserved
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Example: + − k k k x k ( 2) 3 ( 1) 2