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Section10.1-Sequences

# Anotherexample sin2 n

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Unformatted text preview: cannot be applied to a sequence, but can be applied to the corresponding continuous function. Another Example sin(2 n) Is the sequence, , convergent or divergent? Solution: lim sin(2 x) x does not exist, but lim sin(2 n) 0 , so the sequence is convergent. n Theorem 2 lim a 0 If , then lim a n n n n 0 Examples n 1 1 Does the sequence, , converge or n diverge? Solution: 1 n 11 nn 1 n1 0, so lim 1 0 n n n n lim n1 1 converges to 0 n Example r For what values of r does converge? n Solution: 0 for 0 r 1 lim r x 1 for r 1 x for r 1 SO 0 for 0 r 1 convergent n lim r 1 for r 1 convergent n for r 1 divergent Note that rx is defined only for r > 0. What about r ≤ 0 ? Example ‐ continued lim r n 0 if r 1 n lim r n does not exist if r 1 n For example 1 n 1 1 1 1 , , , ,... 2 2 4 8 16 1 1,1, 1,1,... n 2 2, 4, 8,16,... convergent divergent n divergent Summary For {rn} r n r n converges to 0 for 1 r 1 converges to 1 for r = 1 r n diverges otherwise Try It 1 p n For what values of p does converge? Try It...
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