Textbook Question
55–70. More sequences
Find the limit of the following sequences or determine that the sequence diverges.
aₙ = (−1)ⁿ ⁿ√n
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14. Sequences & Series
Sequences
1:47 minutesProblem 10.2.65
Textbook Question
55–70. More sequences
Find the limit of the following sequences or determine that the sequence diverges.
{cosn / n}
Verified step by step guidance1
Identify the given sequence as \( a_n = \frac{\cos n}{n} \), where \( n \) is a positive integer.
Recall that \( \cos n \) oscillates between -1 and 1 for all integer values of \( n \), so the numerator is bounded.
Note that the denominator \( n \) increases without bound as \( n \to \infty \).
Use the Squeeze Theorem by bounding the sequence: since \( -1 \leq \cos n \leq 1 \), we have \( -\frac{1}{n} \leq \frac{\cos n}{n} \leq \frac{1}{n} \).
Since both \( -\frac{1}{n} \) and \( \frac{1}{n} \) approach 0 as \( n \to \infty \), conclude that \( \lim_{n \to \infty} \frac{\cos n}{n} = 0 \).
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Video duration:
1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limit of a Sequence
The limit of a sequence is the value that the terms of the sequence approach as the index n goes to infinity. If the terms get arbitrarily close to a specific number, the sequence converges to that limit; otherwise, it diverges.
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Behavior of the Cosine Function
The cosine function oscillates between -1 and 1 for all real numbers n. Since cos(n) does not approach a single value as n increases, it is important to consider how this oscillation affects the sequence when combined with other terms.
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Squeeze Theorem for Sequences
The Squeeze Theorem states that if a sequence is bounded above and below by two sequences that both converge to the same limit, then the original sequence also converges to that limit. This is useful when dealing with sequences involving bounded oscillating functions divided by terms that grow without bound.
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