Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. Suppose 0 < aₖ < bₖ. If ∑ (k = 1 to ∞) aₖ converges, then ∑ (k = 1 to ∞) bₖ converges.

Verified step by step guidance

1. Step 1: Understand the given inequality and series. We have two sequences \(a_k\) and \(b_k\) such that \(0 < a_k < b_k\) for all \(k\), and the series \(\sum_{k=1}^\infty a_k\) converges.
2. Step 2: Recall the Comparison Test for series convergence. If \(0 \leq a_k \leq b_k\) for all \(k\) and \(\sum b_k\) converges, then \(\sum a_k\) also converges. This test gives a condition for \(a_k\) based on \(b_k\), not the other way around.
3. Step 3: Analyze the statement: it claims that if \(\sum a_k\) converges and \(a_k < b_k\), then \(\sum b_k\) converges. This is the reverse of the Comparison Test and is generally not true.
4. Step 4: To disprove the statement, consider a counterexample. For instance, let \(a_k = \frac{1}{k^2}\) which converges, and \(b_k = \frac{1}{k}\) which is larger than \(a_k\) but whose series diverges.
5. Step 5: Conclude that since \(\sum a_k\) converges but \(\sum b_k\) diverges in this example, the original statement is false.

Key Concepts

Comparison Test for Series

The Comparison Test helps determine the convergence of a series by comparing it to another series with known behavior. If 0 ≤ aₖ ≤ bₖ for all k and ∑ bₖ converges, then ∑ aₖ also converges. However, the converse is not necessarily true.

Convergence of Positive Term Series

A series with positive terms converges if its partial sums approach a finite limit. Understanding that larger terms can cause divergence is key; thus, if a smaller series converges, a larger one may still diverge.

Counterexamples in Series Convergence

Counterexamples demonstrate why a statement is false by providing a specific case that violates it. For series, showing a convergent smaller series and a divergent larger series disproves the claim that convergence of the smaller implies convergence of the larger.