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Further Notes on Sequences

Further Notes on Sequences - Further Notes on Sequences...

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Further Notes on Sequences Additional Methods of Determining Convergence/Divergence of a Sequence : Using rules to evaluate limits : Just as there are rules for evaluating limits of functions, there are similar rules for evaluating limits of sequences. They are listed below. Suppose { } n a and { } n b are convergent sequences and c is a constant. Then 1. ( 29 lim lim lim n n n n n n n a b a b →∞ →∞ →∞ ± = ± 2. ( 29 lim lim lim n n n n n n n a b a b →∞ →∞ →∞ = 3. lim n c c →∞ = ; ( 29 lim lim n n n n ca c a →∞ →∞ = 4. lim lim lim n n n n n n n a a b b →∞ →∞ →∞ = , provided lim 0 n n b →∞ 5. ( 29 lim lim p p n n n n a a →∞ →∞ = 6. If n n a b for all but a finite number of terms, then lim lim n n n n a b →∞ →∞ Item 5 above holds provided the symbols make sense. For example, one cannot take the square root of negative numbers. Example: Below is an example in which this method is used to determine convergence or divergence. E1) Consider 2 1 2 4 n n n = + . 2 2 2 2 2 2 2 2 lim lim lim 4 4 4 1 n n n n n n n n n n n - - →∞ →∞ →∞ = = + + + (Algebra) 2 2 lim 4 lim 1 n n n n →∞ →∞ = + 2 1 2lim 1 lim1 4lim n n n n n →∞ →∞ →∞ = + (#4, and #3, #1) 2 1 2lim 1 lim1 4 lim n n n n n →∞ →∞ →∞ = + (#5)
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( 29 2 2 0 0 1 4 0 = = + . (Evaluating the limits) Comparing with other sequences
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