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)? (a) $\sum_{n=1}^{\infty} \frac{1}{n^{3}}$ (b) $\sum_{n=1}^{\infty} \frac{n}{2^{n}}$ (c) $\sum_{n=1}^{\infty} \frac{(-3)^{n-1}}{\sqrt{n}}$ (d) $\sum_{n=1}^{2} \frac{\sqrt{n}}{1+n^{2}}$

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

Key Concepts

Ratio Test

The Ratio Test is a method for determining the convergence of an infinite series by examining the limit of the ratio of successive terms. Specifically, when you calculate the limit L = lim (|a???/a?|) as n approaches infinity, this test provides clear conclusions: if L < 1, the series converges absolutely; if L > 1, the series diverges. However, if L equals 1, the Ratio Test does not provide any information about the series’ convergence, meaning the test is inconclusive.

Inconclusive Case (Limit = 1)

A key aspect of the Ratio Test is its borderline case: when the limit of the ratio of successive terms is exactly 1, the test fails to offer any insight into whether the series converges or diverges. This is significant because many series, particularly those whose term behavior does not involve dominant exponential growth or decay, may yield a limit of 1. In such cases, alternative convergence tests must be applied.

p-Series

A p-series is an infinite series of the form ? 1/n?, where the parameter p determines its convergence properties. If p > 1, the series converges, while it diverges if p ? 1. Although p-series convergence is well understood through the p-test, applying the Ratio Test to a p-series typically results in a limit of 1, which renders the Ratio Test inconclusive. Thus, recognizing a series as a p-series provides a more direct method to determine convergence when the Ratio Test fails.

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