Unformatted text preview: n →∞  a n  1 /n = L . If L < 1 the series ∑ a n converges absolutely, if L > 1 the series diverges, and if L = 1 this test gives no information. The proof of the root test is actually easier than that of the ratio test, and is a good exercise. EXAMPLE 10.42 Analyze ∞ X n =0 5 n n n . The ratio test turns out to be a bit diﬃcult on this series (try it). Using the root test: lim n →∞ ± 5 n n n ¶ 1 /n = lim n →∞ (5 n ) 1 /n ( n n ) 1 /n = lim n →∞ 5 n = 0 . Since 0 < 1, the series converges. The root test is frequently useful when n appears as an exponent in the general term of the series....
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 Spring '07
 JonathanRogawski
 Math, Calculus, Sequences And Series, Mathematical Series, lim

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