Lecture 22 (Ratio Test and Root Test)

Lecture 22 (Ratio Test and Root Test) - Tuesday February 28...

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Tuesday February 28 Lecture 22 : Ratio test and Root test . (Refers to Section 8.5 in your text) After having practiced the problems associated to the concepts of this lecture the student should be able to : Apply the ratio test and the root test to determine absolute convergence or divergence. We will now discuss two tests for absolute convergence of a given series of numbers These are called the Ratio test and the Root test . It will be important to remember that these do not test for conditional convergence. They only test for “absolute convergence” or “non-absolute convergence” of the original series. 22.1 Theorem Ratio test . (Also called Limit ratio test ). Suppose { a j : j = 1, 2, 3, . .} is a sequence of numbers such that Then Proof the case ρ < 1 and the case ρ > 1 or is given at the end of this lecture. It can be considered as optional reading. It may be omitted without loss of continuity . 22.1.1 Remarks. If ρ > 1, not only does Σ j = 1 to | a j | not converge , neither does Σ j = 1 to a j . So if ρ > 1 the series cannot even converge conditionally. It is useful to retain the following fact:
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If Σ j = 1 to a j converges conditionally ⇒ ρ is not less than 1 and ρ is not greater than 1 or equal to (otherwise this series would diverge) and so the ratio test must fail. If a series converges conditionally, the ratio test will fail.
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This note was uploaded on 04/01/2012 for the course MATH 118 taught by Professor Zhou during the Winter '08 term at Waterloo.

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Lecture 22 (Ratio Test and Root Test) - Tuesday February 28...

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