Power Series (Overview) Bernd Schr oder logo1 Bernd Schr oder - PowerPoint PPT Presentation

Summary Radius of Convergence Representation of Functions Power Series (Overview) Bernd Schr¨ oder logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Power Series (Overview)

Summary Radius of Convergence Representation of Functions Facts About Power Series logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Power Series (Overview)

Summary Radius of Convergence Representation of Functions Facts About Power Series ∞ c n ( x − x 0 ) n is 1. If { c n } ∞ ∑ n = 0 is a sequence of numbers, then n = 0 called a power series about x 0 . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Power Series (Overview)

Summary Radius of Convergence Representation of Functions Facts About Power Series ∞ c n ( x − x 0 ) n is 1. If { c n } ∞ ∑ n = 0 is a sequence of numbers, then n = 0 called a power series about x 0 . 2. There is a nonnegative R (the radius of convergence ), which could be infinity, so that the power series converges for every x in ( x 0 − R , x 0 + R ) and so that the power series diverges for every x not in ( x 0 − R , x 0 + R ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Power Series (Overview)

Summary Radius of Convergence Representation of Functions Facts About Power Series ∞ c n ( x − x 0 ) n is 1. If { c n } ∞ ∑ n = 0 is a sequence of numbers, then n = 0 called a power series about x 0 . 2. There is a nonnegative R (the radius of convergence ), which could be infinity, so that the power series converges for every x in ( x 0 − R , x 0 + R ) and so that the power series diverges for every x not in ( x 0 − R , x 0 + R ) . 3. For R < ∞ , to decide about convergence at x 0 − R and x 0 + R , further convergence tests for series are needed. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Power Series (Overview)

Summary Radius of Convergence Representation of Functions Facts About Power Series ∞ c n ( x − x 0 ) n is 1. If { c n } ∞ ∑ n = 0 is a sequence of numbers, then n = 0 called a power series about x 0 . 2. There is a nonnegative R (the radius of convergence ), which could be infinity, so that the power series converges for every x in ( x 0 − R , x 0 + R ) and so that the power series diverges for every x not in ( x 0 − R , x 0 + R ) . 3. For R < ∞ , to decide about convergence at x 0 − R and x 0 + R , further convergence tests for series are needed. 4. Standard way to compute the radius of convergence for many series: Apply the ratio test . The power series c n + 1 x n + 1 � � � � converges for all x for which lim � < 1 and it � � c n x n n → ∞ � diverges for all x for which the limit is greater than 1. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Power Series (Overview)

Summary Radius of Convergence Representation of Functions Compute the Radius of Convergence of the Power ∞ ( − 2 ) n ∑ n + 1 x n Series n = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Power Series (Overview)

Summary Radius of Convergence Representation of Functions Compute the Radius of Convergence of the Power ∞ ( − 2 ) n ∑ n + 1 x n Series n = 0 ( − 2 ) n + 1 � � ( n + 1 )+ 1 x n + 1 � � � � lim � � ( − 2 ) n n → ∞ n + 1 x n � � � � logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Power Series (Overview)

Summary Radius of Convergence Representation of Functions Compute the Radius of Convergence of the Power ∞ ( − 2 ) n ∑ n + 1 x n Series n = 0 ( − 2 ) n + 1 � � 2 n + 1 ( n + 1 )+ 1 x n + 1 � � n + 2 x n + 1 � � � � � � = lim lim � � � � ( − 2 ) n 2 n � n + 1 x n � n → ∞ n → ∞ n + 1 x n � � � � � � logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Power Series (Overview)

Summary Radius of Convergence Representation of Functions Compute the Radius of Convergence of the Power ∞ ( − 2 ) n ∑ n + 1 x n Series n = 0 ( − 2 ) n + 1 � � 2 n + 1 ( n + 1 )+ 1 x n + 1 � � n + 2 x n + 1 2 n + 1 � � n + 1 � � � � = � = lim 2 n | x | lim lim � � � � ( − 2 ) n 2 n n + 2 � n + 1 x n � n → ∞ n → ∞ n → ∞ n + 1 x n � � � � � logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Power Series (Overview)

Summary Radius of Convergence Representation of Functions Compute the Radius of Convergence of the Power ∞ ( − 2 ) n ∑ n + 1 x n Series n = 0 ( − 2 ) n + 1 � � 2 n + 1 ( n + 1 )+ 1 x n + 1 � � n + 2 x n + 1 2 n + 1 � � n + 1 � � � � = � = lim 2 n | x | lim lim � � � � ( − 2 ) n 2 n n + 2 � n + 1 x n � n → ∞ n → ∞ n → ∞ n + 1 x n � � � � � n → ∞ 2 n + 1 = n + 2 | x | lim logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Power Series (Overview)

Summary Radius of Convergence Representation of Functions Compute the Radius of Convergence of the Power ∞ ( − 2 ) n ∑ n + 1 x n Series n = 0 ( − 2 ) n + 1 � � 2 n + 1 ( n + 1 )+ 1 x n + 1 � � n + 2 x n + 1 2 n + 1 � � n + 1 � � � � = � = lim 2 n | x | lim lim � � � � ( − 2 ) n 2 n n + 2 � n + 1 x n � n → ∞ n → ∞ n → ∞ n + 1 x n � � � � � n → ∞ 2 n + 1 = n + 2 | x | = 2 | x | lim logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Power Series (Overview)

Summary Radius of Convergence Representation of Functions Compute the Radius of Convergence of the Power ∞ ( − 2 ) n ∑ n + 1 x n Series n = 0 ( − 2 ) n + 1 � � 2 n + 1 ( n + 1 )+ 1 x n + 1 � � n + 2 x n + 1 2 n + 1 � � n + 1 � � � � = � = lim 2 n | x | lim lim � � � � ( − 2 ) n 2 n n + 2 � n + 1 x n � n → ∞ n → ∞ n → ∞ n + 1 x n � � � � � n → ∞ 2 n + 1 ! = n + 2 | x | = 2 | x | lim < 1 , logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Power Series (Overview)

Summary Radius of Convergence Representation of Functions Compute the Radius of Convergence of the Power ∞ ( − 2 ) n ∑ n + 1 x n Series n = 0 ( − 2 ) n + 1 � � 2 n + 1 ( n + 1 )+ 1 x n + 1 � � n + 2 x n + 1 2 n + 1 � � n + 1 � � � � = � = lim 2 n | x | lim lim � � � � ( − 2 ) n 2 n n + 2 � n + 1 x n � n → ∞ n → ∞ n → ∞ n + 1 x n � � � � � n → ∞ 2 n + 1 ! = n + 2 | x | = 2 | x | lim < 1 , which is the case exactly when | x | < 1 2. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Power Series (Overview)

Summary Radius of Convergence Representation of Functions Compute the Radius of Convergence of the Power ∞ ( − 2 ) n ∑ n + 1 x n Series n = 0 ( − 2 ) n + 1 � � 2 n + 1 ( n + 1 )+ 1 x n + 1 � � n + 2 x n + 1 2 n + 1 � � n + 1 � � � � = � = lim 2 n | x | lim lim � � � � ( − 2 ) n 2 n n + 2 � n + 1 x n � n → ∞ n → ∞ n → ∞ n + 1 x n � � � � � n → ∞ 2 n + 1 ! = n + 2 | x | = 2 | x | lim < 1 , which is the case exactly when | x | < 1 2. Hence R = 1 2. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Power Series (Overview)

Summary Radius of Convergence Representation of Functions Compute the Radius of Convergence of the Power ∞ 1 ∑ n ! x n Series n = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Power Series (Overview)

Summary Radius of Convergence Representation of Functions Compute the Radius of Convergence of the Power ∞ 1 ∑ n ! x n Series n = 0 ( n + 1 ) ! x n + 1 1 � � � � lim � � 1 n → ∞ � n ! x n � � � logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Power Series (Overview)

Summary Radius of Convergence Representation of Functions Compute the Radius of Convergence of the Power ∞ 1 ∑ n ! x n Series n = 0 ( n + 1 ) ! x n + 1 1 � � n ! � � = ( n + 1 ) ! | x | lim lim � � 1 n → ∞ � n ! x n � n → ∞ � � logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Power Series (Overview)

Summary Radius of Convergence Representation of Functions Compute the Radius of Convergence of the Power ∞ 1 ∑ n ! x n Series n = 0 ( n + 1 ) ! x n + 1 1 � � n ! � � = ( n + 1 ) ! | x | lim lim � � 1 n → ∞ � n ! x n � n → ∞ � � n · ( n − 1 ) ··· 3 · 2 · 1 = ( n + 1 ) · n · ( n − 1 ) ··· 3 · 2 · 1 | x | lim n → ∞ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Power Series (Overview)