L25 Representation of Functions as Power Series I

# L25 Representation of Functions as Power Series I - a 3 x...

This preview shows pages 1–7. Sign up to view the full content.

Lecture 25: Representation of functions as Power Series (I) How do we evaluate such integral Z ln(1 ± t ) t dt ? How do we evaluate such integral Z 0 : 2 0 1 1 + x 5 dx ? Recall the Geometric Series 1 X n =0 x n = 1 1 ± x ; j x j < 1 Using this series, you can represent other functions as power series. 1

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

View Full Document
Express the functions as power series: ex. 3 1 ± x 4 ex. x 4 x + 1 2
ex. 3 x 2 + x ± 2 3

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

View Full Document
Di±erentiation and Integration of Power Series You can di±erentiate and integrate power series term-by-term. The radius of convergence re- mains the same, but the interval of convergence could change. Theorem: If the power series P a n ( x ± a ) n has radius of convergence R > 0, then the func- tion f ( x ) = P 1 n =0 a n ( x ± a ) n = a 0 + a 1 ( x ± a )+ a 2 ( x ± a ) 2 + ²²² is di±erentiable on ( a ± R;a + R ), and 1). f 0 ( x ) = a 1 + 2 a 2 ( x ± a ) + 3

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: a 3 ( x ± a ) 2 + ²²² = 1 X n =1 na n ( x ± a ) n ± 1 2). Z f ( x ) dx = C + a ( x ± a ) + a 1 ( x ± a ) 2 2 + ²²² = C + 1 X n =0 a n ( x ± a ) n +1 n + 1 4 ex. (a). Find the power series representation of the function f ( x ) = ln(1 ± x ) : (b) Approximate ln 2 without a calculator. (excited yet?) 5 Power Series for Arctan x It can easily be shown that 1 1 + x 2 = 1 X n =0 ( ± 1) n x 2 n for ± 1 ² x ² 1 : 6 ex. (1)Find the power series representation for arctan x . (2) Evaluate arctan 1. Tonight’s homework: Write functions as power series. 7...
View Full Document

{[ snackBarMessage ]}

### Page1 / 7

L25 Representation of Functions as Power Series I - a 3 x...

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online