MATH231 Lecture Notes 6

MATH231 Lecture Notes 6 - Root and Ratio tests, Absolute...

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Root and Ratio tests, Absolute Convergence Last time we saw that it is much easier for an alternating series to converge than it is for a series of positive terms. The alternating series test tells us that if a n is positive and decreasing and approaches zero then ( - 1) n a n is convergent. It is often important to be able to decide if a series converges because of “cancellation” between positive and negative terms. This leads to the following concept: Definition: Absolute Convergence A series X a n is absolutely convergent if the series X | a n | is also convergent. A series which is not absolutely convergent (in other words a series for which X a n converges but for which X | a n | diverges is said to be be “conditionally convergent”. Example 1a: X ( - 1) n n 2 ( n + 1) 3 1
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Example 1b: X ( - 1) n n 3 We are going to learn two new convergence tests: the ratio test and the root test. Let me first state the two tests: Theorem: Ratio Test Consider the series X a n and assume that the limit of the ratio exists:
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MATH231 Lecture Notes 6 - Root and Ratio tests, Absolute...

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