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."

Verified step by step guidance

1. 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.
2. 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.
3. 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.
4. 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.
5. 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.

Key Concepts

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.