Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


d. The Ratio Test is always inconclusive when applied to ∑ aₖ, where aₖ is a nonzero rational function of k.

Verified step by step guidance

1. Recall the Ratio Test: For a series \( \sum a_k \), the Ratio Test considers the limit \( L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| \). If \( L < 1 \), the series converges absolutely; if \( L > 1 \), the series diverges; if \( L = 1 \), the test is inconclusive.
2. Understand what it means for \( a_k \) to be a nonzero rational function of \( k \): \( a_k = \frac{p(k)}{q(k)} \), where \( p(k) \) and \( q(k) \) are polynomials and \( a_k \neq 0 \) for all \( k \).
3. Analyze the behavior of the ratio \( \left| \frac{a_{k+1}}{a_k} \right| = \left| \frac{p(k+1)}{q(k+1)} \cdot \frac{q(k)}{p(k)} \right| \). Since \( p(k) \) and \( q(k) \) are polynomials, the ratio of consecutive terms tends to 1 as \( k \to \infty \) because the highest degree terms dominate and their ratio approaches 1.
4. Since the limit \( L \) of the ratio is 1, the Ratio Test is inconclusive for series where \( a_k \) is a nonzero rational function of \( k \).
5. Therefore, the statement is true: the Ratio Test is always inconclusive when applied to \( \sum a_k \) with \( a_k \) a nonzero rational function of \( k \).

Key Concepts

Ratio Test

The Ratio Test is a method to determine the convergence or divergence of an infinite series by examining the limit of the absolute value of the ratio of consecutive terms. If this limit is less than 1, the series converges absolutely; if greater than 1, it diverges; if equal to 1, the test is inconclusive.

Rational Functions of k

A rational function of k is a ratio of two polynomials in the variable k. Such functions often appear in series terms, and their growth rates influence the behavior of the series, especially when applying convergence tests like the Ratio Test.

Limit Behavior of Rational Functions in Series

When applying the Ratio Test to series with terms as rational functions, the limit of the ratio of consecutive terms often approaches 1, making the test inconclusive. Understanding how polynomial degrees in numerator and denominator affect this limit is key to analyzing convergence.