40–62. Choose your test Use the test of your choice to determine whether the following series converge.
∑ (k = 1 to ∞) tan(1 / k)

검증된 단계별 안내

Identify the series given: \( \sum_{k=1}^{\infty} \tan\left(\frac{1}{k}\right) \). We want to determine if this series converges or diverges.

Recall that for large \( k \), \( \frac{1}{k} \) approaches 0, so consider the behavior of \( \tan(x) \) as \( x \to 0 \). Use the fact that \( \tan(x) \approx x \) when \( x \) is close to 0.

Approximate the terms of the series for large \( k \): \( \tan\left(\frac{1}{k}\right) \approx \frac{1}{k} \). This suggests the series behaves similarly to the harmonic series \( \sum \frac{1}{k} \).

Since the harmonic series \( \sum \frac{1}{k} \) diverges, use the Limit Comparison Test with \( a_k = \tan\left(\frac{1}{k}\right) \) and \( b_k = \frac{1}{k} \). Compute \( \lim_{k \to \infty} \frac{a_k}{b_k} \).

If the limit is a finite nonzero number, then both series either converge or diverge together. Since \( \sum \frac{1}{k} \) diverges, conclude the behavior of the original series accordingly.

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주요 개념

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

Convergence of Infinite Series

An infinite series converges if the sequence of its partial sums approaches a finite limit. Understanding convergence is essential to determine whether the sum of infinitely many terms results in a finite value or diverges to infinity or oscillates.

Comparison and Limit Comparison Tests

These tests compare the given series to a known benchmark series to determine convergence. The limit comparison test involves taking the limit of the ratio of terms from two series, helping to conclude convergence or divergence based on the behavior of the simpler series.

Behavior of tan(1/k) for Large k

As k approaches infinity, 1/k approaches zero, and tan(1/k) behaves similarly to 1/k because tan(x) ~ x near zero. Recognizing this helps approximate the series terms and apply appropriate convergence tests by comparing to the harmonic series.

40–62. Choose your test Use the test of your choice to determine whether the following series converge.∑ (k = 1 to ∞) tan(1 / k)