55–70. More sequences
Find the limit of the following sequences or determine that the sequence diverges.

{cosn / n}

검증된 단계별 안내

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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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.

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.

55–70. More sequencesFind the limit of the following sequences or determine that the sequence diverges.{cosn / n}