![]() You are reading notes about how to validate the ratio test.You would have to first derive the facts about geometric series in some other fashion.Īs for your notes, the most likely explanations are: However, if you happen to be in the process of validating the ratio test, it would not be valid to use the ratio test to justify any of the facts you need - such as when geometric series converge - in its validation. It may even be the preferable tool for deciding when a geometric series converges, simply to cut down on the number of things one needs to remember. The ratio test is an example of such a mathematical tool, and is perfectly applicable to geometric series. Designing and validating the mathematical tools used in the above bullet.Using mathematical tools to do calculations and prove things.Two major activities mathematicians (and people who use mathematics) engage in are: We can't know for sure (unless context from the notes you've omitted says something on the topic), but the various comments appearing in this topic suggest an explanation. Basically no computation is needed to show that a geometric series converges, while a couple of computational steps are needed in order to invoke the ratio test. ![]() Moreover, I don't see why one would want to use the ratio test to show that a geometric series converges. This is particularly important in a pedagogical setting, when students may not be entirely cognizant of the line of reasoning that lead up to a result (it is hard to keep track of all of the lemmata, theorems, and proofs that lead up to a result if it is the first time that you have had to deal with them). While one could use the ratio test to establish the convergence of a geometric series (there is nothing stopping us!), it is typically poor style to rely on circular arguments as it can (potentially) lead to overlooking important hypotheses or exceptional cases. Since the proof of the ratio test relies on this convergence, it is circular to argue that a geometric series converges by the ratio test (unless, of course, you have another proof for the ratio test that doesn't use the convergence of geometric series). The key point is that we prove that the ratio test implies the convergence of series by comparison to a convergent geometric series. The other inequality in the ratio test can be argued by noting that the general term does not go to zero, and the uncertainty at 1 can be argued by considering (for example) the harmonic and alternating harmonic series. If $$\lim_ \right| < 1,Ĭonverges (by the limit comparison test, for example). Here’s another example that we can use to understand what makes convergent series special.The usual proof of the ratio test is to compare the series to a geometric series. This means that the sum of a convergent series will approach a certain value as we add more terms and approach infinity. Let’s begin this section by visualizing how terms of convergent series appear on a graph.įrom this, we can see that the series’s partial sums approach a certain number as the value of $n$ increases. Let’s go ahead and first visualize what it means to have a convergent series. We’ll also learn how we can confirm if a given series is convergent or not. In this article, we’ll focus on understanding what makes convergent series unique.
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