MAE101 CAL v2 chapter 8 power series

ĐẠI HỌC FPT CẦN THƠ Chapter Power Series (Page 531-581, Calculus Volume 2) Dr Tran Quoc Duy Email: duytq4@fpt.edu.vn Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Topics : 6.1 Power series and Functions 6.3 Taylor and Maclaurin series Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ  k c x Power series is a series of the form:  k k =0 where x is a variable, and ck are constants called the coefficients of the series Power series centered at a is a series of the form:  k c ( x − a ) k k =0 Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ 6.1 Power series and Functions Power series Example • Geometric series:  k cx  k =0  k c ( x − a ) • The series  k is not called a power series if p< k =p Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ 6.1 Power series and Functions Power series Theorem  Let k c ( x − a ) be a power series centered at a There are  k k =0 only three possibilities: • The series converges only when x= a • The series converges for all x • There is a positive number R such that the series converges if |x-a|R Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ 6.1 Power series and Functions Power series Definition The radius of convergence of the power series  k c ( x − a ) k k =0 is the number R such that the series converges for |x – a| < R and diverges for |x – a| > R Interval of convergence of the power series is the interval that consists of all x for which the series converges Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Example Find the radius and interval of convergence of the series  xk  k =1 k ck +1 k +1 lim = lim = 1→ R = k → c k → k k → the series converges when −1