We have just learned about one of the big and important tools that are used for testing convergence. The Limit Comparison Test is super useful for rational functions and things that act like \(p\)-Series, since these power functions and related function types behave so nicely, and are so friendly to work with, in the end behavior limit. The algebra typically works well, and we can analyze the limits pretty easily.
In this section, we’re going to try to draw a similar connection between our other family of common series, the Geometric Series, and series that act similarly. So for us to begin, we want to think about what it might mean for a series to have terms that act similarly to the terms of a geometric series.
We are going to build some convergence tests that try to link some infinite series to the family of geometric series and show that even though a series is not geometric, it might act enough like one to be considered “eventually geometric-ish.”
Can you describe this characteristic in another way? For instance, if you described a geometric series using a characteristic about the Explicit Formula, can you describe the same characteristic in the context of the Recursion Relation instead? Or vice versa?
Write out a generalized and simplified form of the term \(a_k\) of a geometric series explicitly and recursively. In each case, solve for \(r\text{,}\) the ratio between terms.
We’re going to use these two guiding features of a geometric series to determine if a series is geometric-ish. That is, if a series is not actually a geometric series, do the terms act like terms from a geometric series in the limit? Is there some eventually (almost) constant ratio between consecutive terms? Is there one in the limit as \(k\to\infty\text{?}\) If the terms aren’t actually exponential functions of \(k\text{,}\) do they kind of act like that in the limit as \(k\to\infty\text{?}\) If so, shouldn’t they act like geometric series and converge with the same criteria?
Let \(\displaystyle\sum_{k=0}^\infty a_k\) be an infinite series with \(a_k\gt 0\) for \(k\geq 0\) and consider \(\displaystyle\lim_{k\to\infty} \sqrt[k]{a_k}\text{.}\)
If there is some real number \(r\) with \(\displaystyle r=\lim_{k\to\infty} \sqrt[k]{a_k}\) and \(0\leq r \lt 1\text{,}\) then the series \(\displaystyle\sum_{k=0}^\infty a_k\) converges.
If there is some real number \(r\) with \(\displaystyle r=\lim_{k\to\infty} \sqrt[k]{a_k}\) and \(r\gt\) or if \(\displaystyle\lim_{k\to\infty} \sqrt[k]{a_k}\) does not exist, then the series \(\displaystyle\sum_{k=0}^\infty a_k\) diverges.
Let \(\displaystyle\sum_{k=0}^\infty a_k\) be an infinite series with \(a_k\gt 0\) for \(k\geq 0\) and consider \(\displaystyle\lim_{k\to\infty} \frac{a_{k+1}}{a_k}\text{.}\)
If there is some real number \(r\) with \(r=\displaystyle\lim_{k\to\infty} \frac{a_{k+1}}{a_k}\) and \(0\leq r \lt 1\text{,}\) then the series \(\displaystyle\sum_{k=0}^\infty a_k\) converges.
If there is some real number \(r\) with \(r=\displaystyle\lim_{k\to\infty} \frac{a_{k+1}}{a_k}\) and \(r\gt\) or if \(\displaystyle\lim_{k\to\infty} \frac{a_{k+1}}{a_k}\) does not exist, then the series \(\displaystyle\sum_{k=0}^\infty a_k\) diverges.
So these tests are good and fine, but when do we use them? How do we know that they can be helpful? The key is to notice behavior in the terms of a series that look “kind of” geometric: we’re looking for \(k\) up in the exponent on things and we’re looking for repeated multiplication.
We’re going to look at a couple of small examples where we can rewrite some expressions into friendlier forms, and try to connect these rewriting strategies to the Ratio and Root Tests.
This is an alternating series! We can show that this series converges using the Alternating Series Test, and so we really need to test for absolute convergence.
The Direct Comparison Test is inconclusive when the inequality is the “wrong” way around (like when the series we’re interested in has terms smaller than a diverging series, or has terms larger than a converging series). The Limit Comparison Test has the same issue (where the order of the inequality matters) when the limit we get is either 0 or \(\infty\text{.}\)
In general, this means that we can find an example of a series that converges and an example of a series that diverges, where both give the same result of the test. We know, for example, that every converging series has terms where the limit of the terms is 0, but we have also see examples of series whose terms approach 0 in the limit that diverge. The first example we looked at that did this was the Harmonic Series.
We introduced these convergence tests by thinking about whether or not a series could act like a Geometric Series in the long run (in the limit). And the tests do a great job of uncovering this eventually-geometric-ish behavior! So good, in fact, that (almost) any series that eventually acts kind of like a geometric series will be correctly classified as converging or diverging.
So what, then, does it mean when the Ratio or Root Test is inconclusive? Simply this: it means that the series we’re considering does not act like a geometric series.
Note that this limit does not depend on the exponent, \(p\text{,}\) which means that the limit will be 1 when the \(p\)-series converges and also when the \(p\)-series diverges.
So, in a sense, when we get an inconclusive result from the Ratio or Root Tests, we are learning a lot about our series: it doesn’t act like a geometric series, and that’s likely because it does act like a \(p\)-series! We should try to approach this series again, this time looking for a suitable \(p\)-series to compare it to (whether directly or using the Limit Comparison Test).
For the Ratio Test, we note that if \(\displaystyle\lim_{k\to\infty} \left|\frac{a_{k+1}}{a_k}\right| \lt 1\text{,}\) then the series \(\sum a_k\) converges. Why is this? Explain some general intuition for why the Ratio Test guarantees convergence in this case.
For the Root Test, we note that if \(\displaystyle\lim_{k\to\infty} \sqrt[k]{|a_k|}\lt 1\text{,}\) then the series \(\sum a_k\) converges. Why is this? Explain some general intuition for why the Root Test guarantees convergence in this case.
Note that in both the Ratio and Root Tests above, that we use the absolute value of the terms. Is it possible, then, for the Ratio or Root Test to tell us that a series converges conditionally?
For each of the following infinite series, use any methods we’ve learned about so far to determine whether the series converges or diverges. If it converges and you are able to tell what it converges to, say so and explain. If you cannot tell what it converges to, say so and explain. Clearly state the test or method you use or the type of series.
