Chap. 11 - Summary of Convergence Tests

Chap. 11 - Summary of Convergence Tests - Series...

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Special Series with Known Convergence: “p”-series: The Only Two Types of (non-power) series, in this course, for which we that we can compute exact values: Geometric Series: Telescoping Series: Typical case: In General: Page #1 of 5 1 0 1 if 1, diverges if 1. 1 n n n n a ar ar r r r - = = = = < - ∑ ∑ ( 29 0 ( 1) ( ) (1) (0) (2) (1) m n f n f n S f f f f = + - = - + - (3) (2) ( 1) ( 2) f f f m f m + - + - - - O ( ) ( 1) ( 1) ( ) f m f m f m f m + - - + + - ( 29 ( 29 0 ( 1) ( ) lim = lim ( 1) (0) m m n m f n f n S f m f → ∞ = → ∞ + - = + - ( 29 ( 29 ( 29 ( 29 0 0 0 n 0 0 0 ( ) ( ) , where , are positive integers = ( ) ( ) lim ( 1) ( 2) ( ) ( ) ( 1) ( 1) n m n n f n p f n n p f n p f n f m f m f m p f n f n f n p = → ∞ = + - + - = + + + + + + - + + + + + - K K ( 29 ( 29 0 0 0 = lim ( ) ( ) ( 1) ( 1) m p f m f n f n f n p → ∞ - + + + + + - K 1 converges if 1 and diverges if 1 p p p n
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Convergence Tests: Divergence Test: Alternating Series Test (AST): ( But: If the limit is non-zero, this is the condition for the Divergence Test , and the series will diverge ) The Following Four Tests are for use on Non-Negative series. Each can be used on the positive form of the series, , to demonstrate Absolute Convergence. 1) Comparison Test: 2) Limit Comparison Test: Page #2 of 5 lim 0 then diverges (Notice, this test CANNOT show converges!) n n n n a a a → ∞ ∑ ∑ and converges converges diverges no information diverges diverges converges no inf n n n n n n n n n n n n Given a b non negative series b a a b b b a a b b - ∑ ∑ ∑ ∑ ∑ ∑ ormation
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This note was uploaded on 12/14/2010 for the course MATH 1za3 taught by Professor Ben during the Winter '10 term at Macalester.

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Chap. 11 - Summary of Convergence Tests - Series...

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