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

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