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

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

For which of the following series is the Ratio Test inconclusive (that is, if fails to give a definite answer)?

(a) $ \displaystyle \sum_{n = 1}^{\infty} \frac {1}{n^3} $

(b) $ \displaystyle \sum_{n = 1}^{\infty} \frac {n}{n^2} $

(c) $ \displaystyle \sum_{n = 1}^{\infty} \frac {( - 3)^{n-1}}{\sqrt{n}} $

(d) $ \displaystyle \sum_{n = 1}^{\infty} \frac {\sqrt{n}}{1 + n^2} $

Key Concepts

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.

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.

Transcript

00:01 For each of these series, let's determine whether the ratio test is inconclusive.

00:07 So for part a, let's look at this one first.

00:11 This will be our an.

00:15 So we'd like to look at the limit as n goes to infinity, absolute value, a .n plus 1 over a .n.

00:32 So in our case, here's a .n plus 1.

00:45 And then we'll divide that by a .n.

00:52 Now let's go ahead and multiply out.

00:57 This will give us n cubed over n plus 1 cubed.

01:04 Now if you'd like, you could go ahead and use lopetal's rule.

01:09 In any case, you'll get one when you evaluate this limit and we know that the ratio test is inconclusive when the limit equals 1.

01:24 Since our limit equals 1 in part a, we conclude that the ratio test is inconclusive when the limit equals 1 in part a, is inconclusive for the first series.

01:34 And now let's go on to part b.

01:38 Now this is our a -n.

01:41 And we could even go ahead and do a little cancellation to write this as 1 over n.

01:46 So we'll have the limit.

01:49 N goes to infinity.

01:52 Once again, a -n plus 1 over a -n.

02:00 Now the numerator, this is 1 over n plus 1.

02:04 And the denominator, 1 over n.

02:06 Let's go ahead and flip that denominator and then multiply.

02:15 Once again, you could use lopatow's rule here, and you'll get a limit of one, just like in part a, and we conclude that the ratio test is inconclusive here as well.

02:31 So that's for part b...

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