For which of the following series is the Ratio Test...

For which of the following series is the Ratio Test inconclusive (that is, it fails to give a definite answer)? $$\begin{array}{ll}{\text { (a) } \sum_{n=1}^{\infty} \frac{1}{n^{3}}} & {\text { (b) } \sum_{n=1}^{\infty} \frac{n}{2^{n}}} \\ {\text { (c) } \sum_{n=1}^{\infty} \frac{(-3)^{n-1}}{\sqrt{n}}} & {\text { (d) } \sum_{n=1}^{\infty} \frac{\sqrt{n}}{1+n^{2}}}\end{array}$$

For which of the following series is the Ratio Test inconclusive
(that is, it fails to give a definite answer)?
$$\begin{array}{ll}{\text { (a) } \sum_{n=1}^{\infty} \frac{1}{n^{3}}} & {\text { (b) } \sum_{n=1}^{\infty} \frac{n}{2^{n}}} \\ {\text { (c) } \sum_{n=1}^{\infty} \frac{(-3)^{n-1}}{\sqrt{n}}} & {\text { (d) } \sum_{n=1}^{\infty} \frac{\sqrt{n}}{1+n^{2}}}\end{array}$$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Growth Rates in Series Terms

Understanding the behavior of terms in a series—especially the relative growth of polynomial, exponential, or factorial components—is key to applying convergence tests effectively. When exponential factors dominate the terms, the ratio often yields a limit less than one, indicating convergence. Conversely, when the terms involve polynomial expressions, the consecutive term ratios tend to approach 1, rendering the Ratio Test insufficient and inconclusive.

Ratio Test

The Ratio Test is a standard method in the study of series convergence. It involves taking the limit L = lim??? |a??? / a?|, where a? is the nth term of the series. If L < 1, the series converges absolutely; if L > 1, the series diverges; and if L = 1, the test is inconclusive because it does not provide sufficient information to determine convergence or divergence.

Inconclusive Cases of the Ratio Test

The Ratio Test becomes inconclusive when the limit L equals exactly 1. This situation does not allow any definitive conclusion about the convergence of the series and therefore other convergence tests must be employed. Many series with polynomial or slowly varying terms produce a limit of 1 when the Ratio Test is applied, highlighting a common shortcoming of the test in such instances.

Transcript

Please give Ace some feedback