lec36 - .01 Fall 2006 Lecture 36 Infinite Series and...

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Unformatted text preview: Lecture 36 18.01 Fall 2006 Lecture 36: Infinite Series and Convergence Tests Infinite Series Geometric Series A geometric series looks like 1 + a + a 2 + a 3 + ... = S There’s a trick to evaluate this: multiply both sides by a : a + a 2 + a 3 + ... = aS Subtracting, (1 + a + a 2 + a 3 + )- ( a + a 2 + a 3 + ) = S- aS · · · · · · In other words, 1 1 = S- aS = ⇒ 1 = (1- a ) S = ⇒ S = 1- a This only works when | a | < 1 , i.e.- 1 < a < 1 . a = 1 can’t work: 1 + 1 + 1 + ... = ∞ a =- 1 can’t work, either: 1 1 1- 1 + 1- 1 + ... = 1- (- 1) = 2 Notation Here is some notation that’s useful for dealing with series or sums. An infinite sum is written: ∞ a k = a + a 1 + a 2 + ... k =0 The finite sum n S n = a k = a + ... + a n k =0 is called the “ n th partial sum” of the infinite series. 1 Lecture 36 18.01 Fall 2006 Definition ∞ a k = s k =0 means the same thing as n lim S n = s, where S n = a k n →∞ k =0 We say the series converges to s , if the limit exists and is finite. The importance of convergence is illustrated here by the example of the geometric series. If a = 1 , S = 1 + 1 + 1 + ... = ∞ . But S- aS = 1 or ∞-∞ = 1 does not make sense and is not usable! Another type of series: ∞ 1 n p n =1 We can use integrals to decide if this type of series converges. First, turn the sum into an integral: ∞ 1 ∞ dx...
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This note was uploaded on 02/27/2009 for the course MATH 155b taught by Professor Staff during the Fall '08 term at Vanderbilt.

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lec36 - .01 Fall 2006 Lecture 36 Infinite Series and...

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