11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.

∑ (from k = 1 to ∞)(cos(1 / k) – cos(1 / (k + 1)))

검증된 단계별 안내

Recognize that the given series is \( \sum_{k=1}^{\infty} \left( \cos\left(\frac{1}{k}\right) - \cos\left(\frac{1}{k+1}\right) \right) \), which is a telescoping series because each term involves the difference of cosine values at consecutive indices.

Write out the first few partial sums explicitly to observe the telescoping pattern: \( S_n = \sum_{k=1}^n \left( \cos\left(\frac{1}{k}\right) - \cos\left(\frac{1}{k+1}\right) \right) = \left( \cos(1) - \cos\left(\frac{1}{2}\right) \right) + \left( \cos\left(\frac{1}{2}\right) - \cos\left(\frac{1}{3}\right) \right) + \cdots + \left( \cos\left(\frac{1}{n}\right) - \cos\left(\frac{1}{n+1}\right) \right) \).

Notice that most terms cancel out in the sum, leaving only the first term of the first cosine and the last term of the last cosine: \( S_n = \cos(1) - \cos\left(\frac{1}{n+1}\right) \).

To determine convergence, analyze the limit of the partial sums as \( n \to \infty \): \( \lim_{n \to \infty} S_n = \lim_{n \to \infty} \left( \cos(1) - \cos\left(\frac{1}{n+1}\right) \right) \).

Since \( \lim_{x \to 0} \cos(x) = 1 \), evaluate the limit to find \( \lim_{n \to \infty} S_n = \cos(1) - 1 \). This limit exists and is finite, so the series converges by the telescoping series test.

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주요 개념

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

Telescoping Series

A telescoping series is one where many terms cancel out when the series is expanded, leaving only a few terms to evaluate the sum. Recognizing this pattern helps simplify the series and determine convergence by examining the behavior of the remaining terms.

Limit of a Sequence

The limit of a sequence is the value that the terms of the sequence approach as the index goes to infinity. For series convergence, understanding the limit of the partial sums or the terms themselves is crucial to determine if the series converges to a finite value.

Convergence Tests for Series

Convergence tests, such as the comparison test, ratio test, or root test, help determine whether an infinite series converges or diverges. In this problem, identifying the appropriate test or recognizing the telescoping nature is key to justifying convergence.

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.∑ (from k = 1 to ∞)(cos(1 / k) – cos(1 / (k + 1)))