Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
f. If lim (k → ∞) aₖ = 0, then ∑ aₖ converges."

검증된 단계별 안내

Step 1: Understand the statement. It says: If the limit of the sequence \( a_k \) as \( k \to \infty \) is zero, then the infinite series \( \sum a_k \) converges. We need to determine if this is true or false.

Step 2: Recall the necessary condition for series convergence: For a series \( \sum a_k \) to converge, the terms \( a_k \) must approach zero as \( k \to \infty \). This means \( \lim_{k \to \infty} a_k = 0 \) is necessary but not sufficient for convergence.

Step 3: To test if the statement is true, consider a counterexample where \( \lim_{k \to \infty} a_k = 0 \) but \( \sum a_k \) diverges. A classic example is the harmonic series \( a_k = \frac{1}{k} \), which tends to zero but the series \( \sum \frac{1}{k} \) diverges.

Step 4: Explain why the harmonic series diverges despite its terms tending to zero. This shows that having terms approach zero does not guarantee the sum converges.

Step 5: Conclude that the statement is false because the limit of the terms being zero is necessary but not sufficient for the convergence of the series.

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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

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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 {aₖ} as k approaches infinity is the value that the terms of the sequence get arbitrarily close to. If lim (k → ∞) aₖ = 0, it means the terms become very small, but this alone does not guarantee the sum of the terms converges.

Convergence of an Infinite Series

An infinite series ∑ aₖ converges if the sequence of its partial sums approaches a finite limit. Even if the terms aₖ approach zero, the series may diverge if the partial sums grow without bound.

Counterexample: Harmonic Series

The harmonic series ∑ 1/k is a classic example where the terms approach zero, but the series diverges. This shows that having terms tend to zero is necessary but not sufficient for series convergence.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.f. If lim (k → ∞) aₖ = 0, then ∑ aₖ converges."