Section10.1-Sequences

2n 1 3 n 5 2n 1 3 5 2 3 n

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Unformatted text preview: Show whether the following sequences converge or diverge. 2n 1 3 n 5 2n 1 3 5 2 3 n 5 Try It Show whether the following sequences converge or diverge. n n 1 n 1 1 n 1 n n n n 1 n 1 Try It Show whether the following sequences converge or diverge. n ln e 4 3n 1 2 n 1 1n n n 1 n Monotonic Theorem A monotonic bounded sequence converges. an 1 an (monotonically increasing: for all n monotonically decreasing: for all n) an 1 an Explanation Monotonically increasing sequence is always bounded below. Monotonically decreasing sequence is always bounded above. Example Does the sequence, { 0.2, 0.22, 0.222, 0.2222, …} converge? Solution: Sequence is monotonically increasing. It is bounded by 0.2 (below) and 0.3 (above). The sequence is convergent. Squeeze Theorem lim an lim cn L an bn cn n n0 If for all , and , then n n lim bn L n Review Factorial 0!=1 n!=n(n‐1)! 5! 5 4 3 2 1 67! 67 66! n 2 ! n 2 n 1 n ! Example 4n n! Does the sequence, , converge? Solution: nth term is 4 4 4 4 4 4 4 1 2 3 4 5 (n 1) n 4 4 4 4 4 4 4 4 4 4 4 4 for n 5 1 2 3 4 5 (n 1) n 1 2 3 4 n 4n 1024 0 for n 5 n ! 24n 4n lim 0 n n ! AND 1024 128 1 lim 0 n 24n n 3 n lim by the squeeze theorem...
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