lec36 - MIT OpenCourseWare http/ocw.mit.edu 18.01 Single...

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MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus Fall 2006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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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 0 + a 1 + a 2 + ... k =0 The finite sum n S n = a k = a 0 + ... + a n k =0 is called the “ n th partial sum” of the infinite series. 1
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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!
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This note was uploaded on 01/18/2012 for the course MATH 18.01 taught by Professor Brubaker during the Fall '08 term at MIT.

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lec36 - MIT OpenCourseWare http/ocw.mit.edu 18.01 Single...

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