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

Info iconThis preview shows pages 1–3. 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
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

This note was uploaded on 02/27/2009 for the course MATH 155b taught by Professor Staff during the Fall '08 term at Vanderbilt.

Page1 / 7

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

This preview shows document pages 1 - 3. Sign up to view the full document.

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