Explain why or why not
Determine whether the following statements are true and give an explanation or counterexample.
a. If the Limit Comparison Test can be applied successfully to a given series with a certain comparison series, the Comparison Test also works with the same comparison series.

Verified step by step guidance

1. Recall the definitions of the Comparison Test and the Limit Comparison Test for series convergence: The Comparison Test requires that the terms of the given series and the comparison series satisfy an inequality for all sufficiently large indices, while the Limit Comparison Test uses the limit of the ratio of their terms as the index approaches infinity.
2. For the Comparison Test, you need to find a comparison series \( \sum b_n \) such that either \( 0 \leq a_n \leq b_n \) or \( 0 \leq b_n \leq a_n \) for all \( n \) beyond some index, and then use the known convergence or divergence of \( \sum b_n \) to conclude about \( \sum a_n \).
3. For the Limit Comparison Test, you compute \( \lim_{n \to \infty} \frac{a_n}{b_n} = c \), where \( c \) is a finite positive number. If this limit exists and is positive and finite, then both series either converge or diverge together.
4. Consider that the Limit Comparison Test can succeed even if the inequality required for the Comparison Test does not hold for all large \( n \). This means the Comparison Test might fail to apply with the same comparison series even though the Limit Comparison Test works.
5. Therefore, the statement is false: the successful application of the Limit Comparison Test with a certain comparison series does not guarantee that the Comparison Test will also work with the same comparison series. A counterexample can be constructed where the limit of the ratio exists and is positive, but the inequality needed for the Comparison Test fails.

Key Concepts

Limit Comparison Test

The Limit Comparison Test determines the convergence or divergence of a series by comparing it to a second series with known behavior. It involves taking the limit of the ratio of their terms; if the limit is a positive finite number, both series either converge or diverge together.

Comparison Test

The Comparison Test assesses a series' convergence by directly comparing its terms to those of a known convergent or divergent series. If the terms of the original series are smaller than a convergent series, it converges; if larger than a divergent series, it diverges.

Differences Between Limit Comparison and Comparison Tests

While both tests compare series, the Comparison Test requires strict inequalities between terms, which can be restrictive. The Limit Comparison Test is more flexible, relying on the limit of term ratios, so it can succeed even when direct inequalities needed for the Comparison Test do not hold.