1. Let an = nsin(1/n) + (-1)n(cosn/n) for n = 1,2,…. Which statement is true of the sequence {an}?1. Let an = nsin(1/n) + (-1)n(cosn/n) for n = 1,2,…. Which statement is true of the sequence {an}?
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A. It is bounded but does not converge. |
B. It converges to 0. |
C. It converges to a positive number. |
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D. It diverges to infinity. |
E. It is unbounded and contains both arbitrarily large positive and arbitrarily large negative terms. |
Solution: The answer is C
Consider the following limit,
By using the limit laws for sequences, we can rewrite the limit as such,
Applying l’Hospital’s Rule to the first limit above yields,
Since,
Applying the squeeze theorem for sequences yields,
So, the limiting value of an as n goes to infinity is 1;, by definition, the sequence an converges to a positive number.