Lecture46-rev

Lecture46-rev - MAT A26 Lecture 46 1 Differentiating and...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MAT A26 Lecture 46 1 Differentiating and Integrating Power Se- ries proposition 1.1. Let X n =0 a n x n be a real power series with radius of con- vergence R , defining a function f ( x ) for | x | < R . Then (a) the power series X n =0 na n x n- 1 obtained by differentiating the original series term by term also has radius of convergence R and sums to f ( x ). (b) the power series X n =0 a n x n +1 n + 1 obtained by integrating the original series term by term also has radius of convergence R and sums to R x f ( x ) dx . (This is also true for complex power series, although we have never yet defined the derivative and integral for complex functions. This is another cliffhanger!! It will be covered in the course Complex Variables as well. ) example 1.2. Integrating the equation 1 1- x = X n =0 x n (for | x | < 1) yields Z x dx 1- x = X n =0 x n +1 n + 1 = X n =1 x n n Now Z x dx 1- x =- ln(1- x ) | x =- ln(1- x ), so ln(1- x ) =- X n =1 x n n for | x | < 1 1 remark...
View Full Document

This note was uploaded on 03/04/2012 for the course MATH 10250 taught by Professor Himonas during the Fall '08 term at Notre Dame.

Page1 / 3

Lecture46-rev - MAT A26 Lecture 46 1 Differentiating and...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online