lec21 - Math 1 B Power series Jakub Kominiarczuk October 19...

Info iconThis preview shows pages 1–14. 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

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

View Full DocumentRight Arrow Icon

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

View Full DocumentRight Arrow Icon

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

View Full DocumentRight Arrow Icon

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

View Full DocumentRight Arrow Icon

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

View Full DocumentRight 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: Math 1 B: Power series Jakub Kominiarczuk October 19 , 2009 Previously, on Math 1 B.. Testing whether series converge and finding their limits. ∞ ∑ n = 1 1 + 1 n n 2 ( root test, diverges ) ∞ ∑ n = ( 2 / 3 ) n ( ratio test, converges ) So far, each term of the series depended only on n . Previously, on Math 1 B.. Testing whether series converge and finding their limits. ∞ ∑ n = 1 1 + 1 n n 2 ( root test, diverges ) ∞ ∑ n = ( 2 / 3 ) n ( ratio test, converges ) So far, each term of the series depended only on n . Previously, on Math 1 B.. Testing whether series converge and finding their limits. ∞ ∑ n = 1 1 + 1 n n 2 ( root test, diverges ) ∞ ∑ n = ( 2 / 3 ) n ( ratio test, converges ) So far, each term of the series depended only on n . Previously, on Math 1 B.. Testing whether series converge and finding their limits. ∞ ∑ n = 1 1 + 1 n n 2 ( root test, diverges ) ∞ ∑ n = ( 2 / 3 ) n ( ratio test, converges ) So far, each term of the series depended only on n . Previously, on Math 1 B.. Testing whether series converge and finding their limits. ∞ ∑ n = 1 1 + 1 n n 2 ( root test, diverges ) ∞ ∑ n = ( 2 / 3 ) n ( ratio test, converges ) So far, each term of the series depended only on n . Previously, on Math 1 B.. Testing whether series converge and finding their limits. ∞ ∑ n = 1 1 + 1 n n 2 ( root test, diverges ) ∞ ∑ n = ( 2 / 3 ) n ( ratio test, converges ) So far, each term of the series depended only on n . The geometric series The last series ∞ ∑ n = ( 2 / 3 ) n = 3 is the familiar geometric series. In general, it could be written as ∞ ∑ n = r n = 1 1- r for- 1 < r < 1 , with the above example being the case with r = 2 / 3 . The geometric series The last series ∞ ∑ n = ( 2 / 3 ) n = 3 is the familiar geometric series. In general, it could be written as ∞ ∑ n = r n = 1 1- r for- 1 < r < 1 , with the above example being the case with r = 2 / 3 . Introducing x Consider renaming the value r and calling it x instead. ∞ ∑ n = x n . This clearly does not change anything about the series, but introduces a variable . We could define a function f ( x ) to be the sum of the series f ( x ) ≡ ∞ ∑ n = x n = 1 + x + x 2 + x 3 + x 4 + . . . Introducing x Consider renaming the value r and calling it x instead. ∞ ∑ n = x n . This clearly does not change anything about the series, but introduces a variable . We could define a function f ( x ) to be the sum of the series f ( x ) ≡ ∞ ∑ n = x n = 1 + x + x 2 + x 3 + x 4 + . . . Power series The function f ( x ) is an example of a power series , ∞ ∑ n = c n x n = c + c 1 x + c 2 x 2 + . . . + c n x n + . . . Example. Letting c n = 1 for all n gives us the geometric series, ∞ ∑ n = x n = 1 + x + x 2 + . . . + x n + . . . Power series The function f ( x ) is an example of a power series , ∞ ∑ n = c n x n = c + c 1 x + c 2 x 2 + . . . + c n x n + . . ....
View Full Document

This note was uploaded on 01/02/2011 for the course MATH 1B taught by Professor Reshetiken during the Spring '08 term at Berkeley.

Page1 / 59

lec21 - Math 1 B Power series Jakub Kominiarczuk October 19...

This preview shows document pages 1 - 14. Sign up to view the full document.

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