![]() And this should be taken as a warning that while the growth of the logarithm is slow, it cannot be ignored. So, testing by ignoring the logarithm is inconclusive. But, and and doesn’t reveal that the series is divergent. If you look at the criteria of case 3 in the statement, we can only make a conclusion if the series in the denominator diverges. Here’s what happens if we ignore the logarithm: But we can’t simply ignore the logarithm in the series and replace it with one. We can formalize this idea by running LCT. But what’s more interesting is that the numerator grows slower than n 2. In this example, the denominator grows at the rate of n 2. The LCT will likely be in case 2 or case 3 of the statement. When you do this, the logarithmic growth in the series is gone. Since there is a √ n 7, the dominant behavior in the denominator comes from √ n 7. Now, looking at the denominator, you will find three terms under the square root. Since sin n is very different from sin (1/n), we are not in case 3 mentioned in the statement of LCT. And finally, the third term sin n bounces between the interval of -1 and 1 depending on the value of n. The first term, n 2, is the most important term because n grows to infinity in this series. If a series has dominant terms that are apparent when you first look at the series, it is possible to generate a b n to compare the series with, by isolating the series’ dominant behavior.Īt first glance, there’s a lot you can ignore in the series. The LCT is the most useful and most straightforward method to isolate the dominant behavior in a series. There’s a fourth, broad scenario where the LCT can be helpful too. There are three primary scenarios you can apply the limit comparison test in. By the same token, since you know that converges, computing the value doesn’t help you conclude anything. ![]() If you compute that = ∞, you’re essentially arguing that. In the same way, if you pick a series to compare with, and you know that it converges. But since you know that diverges, computing the value doesn’t help you conclude anything. Now, if you compute that = 0, you’re essentially arguing that. Let’s assume that you pick a series to compare with, and you know that the series diverges. Now that you know the statement, it is important to understand the subtleties of cases 2 and 3 above. If = ∞, and diverges, then also diverges (given that ).If = 0, and converges, then also converges (given that ).If is positive and finite, then and either converge or diverge (given that ). ![]() Let’s also assume that there’s a positive sequence b n that is also positive or zero. Let’s assume is an infinite series, where ɑ n is zero or a positive number. In this guide, we take you through the heuristics and subtleties of the limit comparison test to help you use it without a struggle. While very useful and relatively easy to apply, it can be challenging to understand when to use the convergence test and how to use it effectively.
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