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3-Ratio, Root Tests

# 3-Ratio, Root Tests - Ratio and Root Tests John E Gilbert...

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Ratio and Root Tests John E. Gilbert, Heather Van Ligten, and Benni Goetz Properties of geometric series enable us to test whether other series converge or diverge - even if they’re not geometric series! Root Test: an infinite series n a n Converges Absolutely when lim n →∞ | a n | 1 /n < 1 , Diverges when lim n →∞ | a n | 1 /n > 1 , if lim n →∞ | a n | 1 /n = 1 , the test is inconclusive, it tells us nothing. Example 1: does the series n = 1 ( - 1) n - 1 n 2 n + 1 n converge absolutely ? Solution: the Root Test works well here because of the n th -power exponent in a n = ( - 1) n - 1 n 2 n + 1 n . For then | a n | 1 /n = n 2 n + 1 n 1 /n = n 2 n + 1 1 2 as n → ∞ . So the series converges absolutely by the Root Test. What’s the connection with geometric series? Well, lim n → ∞ | a n | 1 /n = L = | a n | L n for all large n, so n | a n | n L n . But we know that the geometric series n L n converges when L < 1 and diverges when L > 1 . With more care, this establishes the Root test.

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There’s another test similar to the Root Test, but one which often works well. It’s connection with
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3-Ratio, Root Tests - Ratio and Root Tests John E Gilbert...

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