Alternating Series Test

Alternating Series Test - A Caution on the Alternating...

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A Caution on the Alternating Series Test Theorem 14 (The Alternating Series Test) of the textbook says: The series X n =1 ( - 1) n +1 u n = u 1 - u 2 + u 3 - u 4 + ··· converges if all of the following conditions are satisfied: 1. u n > 0 for all n N . 2. u n u n +1 for all n N , for some integer N . 3. u n 0 as n → ∞ . Clearly, to show the convergence, you need to check all these three conditions. What can you conclude if a given alternating series fails one or more of the three conditions? Is that series divergent? Unfortunately, the failure of this test does not immediately lead to the divergence! You need to use some other test, often, the n th Term Test for Divergence to conclude that the series diverges. The reason is the following. The equivalent statement to Theorem 14 is that: (1) “If a given alternating series diverges, then it fails to satisfy one or more of the above three conditions.” This is different from the following statement, which is false : (2)
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Alternating Series Test - A Caution on the Alternating...

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