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

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Key Concepts

Ratio Test

The Ratio Test is a method used to determine the convergence or divergence of an infinite series. It involves taking the limit of the absolute value of the ratio of consecutive terms. If this limit is less than one, the series converges absolutely; if it is greater than one, the series diverges; and if it equals one, the test is inconclusive and does not provide definitive information about the convergence of the series.

Inconclusiveness When the Limit Equals One

A key aspect of the Ratio Test is that it fails to yield a conclusion when the limit of the ratio of successive terms equals exactly one. This scenario occurs in many series where the dominant behavior of the numerator and denominator cancel out, such as in series whose terms decay like a polynomial. When the limit is one, other convergence tests must be applied to determine the behavior of the series.

p-Series

A p-series is a series of the form ? 1/n^p where p is a positive constant. These series converge if and only if p > 1. Although p-series are well understood via the p-test, when the Ratio Test is applied to them the resulting limit is always 1, making the Ratio Test inconclusive in such cases. This example highlights the importance of choosing the appropriate convergence test based on the structure of the series.

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