n_13533 - 1 lim n →∞ n b a n = 0 for a> 1 and any b 2...

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Calculus 2: Some Results on Limits of Sequences With limits of sequences, one very important situation is determining when a sequence has limit zero, which is necessary for the sum of the sequence to converge, though not enough to guarantee convergence of the sum. it is also often useful to comapare how fast diFerent sequences go to zero, or to infinity, by looking at the ratio of their terms. Many cases involve powers of n (“polynomial”), n -th powers (“geometric”) and fac- torials, and the following results wil often be useful. One overall pattern to note is that geometric (or exponential) growth a n is faster that power law (polynomial) growth n b , and factorial growth n ! is even faster than geometric.
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Unformatted text preview: 1. lim n →∞ n b a n = 0, for a > 1 and any b . 2. lim n →∞ a n n ! = 0, for any a . 3. lim n →∞ n ! n n = 0. 4. lim n →∞ n + 1 n = 1; more generally 5. lim n →∞ ( n + 1) b n b = 1 for any power b ; and more generally still 6. lim n →∞ p ( n + 1) p ( n ) = 1 for any polynomial p ( n ). 7. lim n →∞ a n +1 a n = a n +1 a n = a (geometric). 8. ( n + 1)! n ! = n + 1, so 9. lim n →∞ ( n + 1)! n ! = ∞ (factorial). 10. lim n →∞ n 1 /n = 1 and more generally 11. lim n →∞ | p ( n ) | 1 /n = 1 for any polynomial p ( n ). 12. lim n →∞ | a n | 1 /n = | a | (geometric). 13. lim n →∞ ( n !) 1 /n = ∞ (factorial)....
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