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AlternatingSeries - Alternating Series John E Gilbert...

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Alternating Series John E. Gilbert, Heather Van Ligten, and Benni Goetz When some terms of an infinite series take positive values, while others take negative values, the convergence is often improved. For instance, when (i) 1 + 1 2 + 1 3 + 1 4 + ... + 1 n + ... , (ii) 1 - 1 2 + 1 3 - 1 4 + ... + ( - 1) n 1 n + ... , the first series diverges, yet the second series converges because adding and subtracting provides cancellation . Let’s get an idea why: grouping the terms in pairs gives parenleftbigg 1 - 1 2 parenrightbigg + parenleftbigg 1 3 - 1 4 parenrightbigg + ... + parenleftbigg 1 2 n - 1 2 n + 1 parenrightbigg + ... = 1 1 . 2 + 1 3 . 4 + ... + 1 2 n (2 n + 1) + ... . But summationdisplay n 1 n 2 converges, while lim n → ∞ n 2 parenleftBigg 1 2 n (2 n + 1) parenrightBigg = 1 4 , so by the Limit Comparison test, 1 + 1 2 2 + ... + 1 n 2 + ... converges = 1 1 . 2 + 1 3 . 4 + ... + 1 2 n (2 n + 1) + ... converges . This can easily be turned into a proof showing that series (ii) converges. Series (ii) is called the Alternating Harmonic Series . Its structure and properties lead to Definition: a series is said to be an Alternating Series when it has the form summationdisplay n ( - 1) n 1 a n with a n > 0 , i.e., when it can be written a 1 - a 2 + a 3 - a 4 + ... + ( - 1) n a n + ... , a n > 0 , or, in other words, when consecutive terms alternate in sign.
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There’s a very useful test for determining whether an alternating series converges: Alternating Series Test: an infinite series summationdisplay n ( - 1) n 1 a n with a n > 0 converges if a n a n +1 for all n N , lim n → ∞ a n = 0 .
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