This is a note written for my Fall 2016 Math 100 class at Brown University. We are currently learning about various tests for determining whether series converge or diverge. In this note, we collect these tests together in a single document. We give a brief description of each test, some indicators of when each test would be good to use, and give a prototypical example for each. Note that we do justify any of these tests here — we've discussed that extensively in class. [But if something is unclear, send me an email or head to my office hours]. This is here to remind us of the variety of the various tests of convergence.
A copy of just the statements of the tests, put together, can be found here. A pdf copy of this whole post can be found here.
In order, we discuss the following tests:
- The
th term test, also called the basic divergence test - Recognizing an alternating series
- Recognizing a geometric series
- Recognizing a telescoping series
- The Integral Test
- P-series
- Direct (or basic) comparison
- Limit comparison
- The ratio test
- The root test
The th term test
Statement
Suppose we are looking at
When to use it
When applicable, the
Example
Each of the series
Recognizing alternating series
Statement
Suppose
, is decreasing, and .
Then
Stated differently, if the terms are alternating sign, decreasing in absolute size, and converging to zero, then the series converges.
When to use it
The key is in the name — if the series is alternating, then this is the goto idea of analysis. Note that if the terms of a series are alternating and decreasing, but the terms do not go to zero, then the series diverges by the
Example
Suppose we are looking at the series
Recognizing geometric series
A geometric series is a series of the from
Statement
Given a geometric series
Further, if
When to use it
(At the risk of pointing out the obvious): Given a geometric series, one should always interpret its convergence by considering the ratio.
Example
Suppose we are considering the series
Recognizing a telescoping series
A series is said to telescope if, after some point, all the terms in the series cancel with later terms in the series. These series are often easier to recognize after writing out several terms in the series (and perhaps after performing a partial fraction decomposition).
Note that telescoping series are some of the few series that you can actually evaluate exactly (when they converge). And note that not every telescoping series converges!
Example
Suppose we are considering the series
The Integral Test
An infinite sum is used both in integrals and in infinite series. The idea of the integral test is that for a nice function
Statement
Suppose that
When to use it
If you recognize a function that you can integrate, then the integral test is very useful. In particular, if you see a function and its derivative (for use in
Example
Suppose we are examining the series
P-series
P-series concern the behavior of the series
Statement
The series
When to use it
On its own, it's only occasionally useful. But its power comes when you use this as a basis for comparison in the Direct Comparison or Limit Comparison tests.
Example
The series
[Note that in this series, the individual terms go to zero and yet the series still diverges!]
Direct comparison
For the majority of interesting series, it is often easier to compare to other, simpler-to-understand series. In particular, it is usually easier to identify either
- a larger convergent series, or
- a smaller divergent series.
Statement
Suppose we are considering the two series
Further, if
This can be restated in the following informal way: if the bigger one converges, then so does the smaller. And in the other direction, if the smaller one diverges, then so does the larger.
When to use it
The two comparison tests don't have clear times to use them. But the idea is this: once you have a suspicion that a series converges or diverges, it is often a good idea to try to simplify the expressions for the terms by bounding them (in the correct direction!).
Some terms that are often good to bound are trigonometric terms (like bounding
Example
Suppose we are considering the series
Together, these mean that
Limit comparison
For most series, the first step is to consider what is looks like for large
Statement
Suppose we are considering the series
[Recall that we discussed a stronger version of this statement in class, concerning what can be said when
When to use it
Limit comparison is a very powerful tool that can be used to remove a lot of the unimportant and smaller parts of terms of a series.
Example
The classic example is to handle gross ratios of polynomials. Suppose we are considering the series
We perform a limit comparison test, comparing our series against the series
Since the limit exists and is not equal to
The ratio test
The ratio test is built on the idea that if the
Statement
Suppose we are considering
If
When to use it
If you see factorials, the ratio test is probably a good thing to use. (It is also very useful when finding the radius of convergence of a power series.)
Example
Suppose that we are considering
Since the limit is
The root test
The root test is based on similar intuition behind the ratio test: if the
Statement
Suppose that we are considering
If the limit does not exist, or if the limit exists and
When to use it
If everything (or almost everything) is raised to the
Example
Suppose that we are considering
Since
Concluding Remarks
This is an overview of each test that we have learned thus far. Note that sometimes, when confronted with a new series, the first technique that you try won't work out. And that's ok! It may be necessary to try a few techniques, or perhaps even combine various tests together in order to understand the convergence or divergence of a series.
If there are any questions, let me know. Good luck, and I'll see you in class.
Leave a comment
Info on how to comment
To make a comment, please send an email using the button below. Your email address won't be shared (unless you include it in the body of your comment). If you don't want your real name to be used next to your comment, please specify the name you would like to use. If you want your name to link to a particular url, include that as well.
bold, italics, and plain text are allowed in
comments. A reasonable subset of markdown is supported, including lists,
links, and fenced code blocks. In addition, math can be formatted using
$(inline math)$
or $$(your display equation)$$
.
Please use plaintext email when commenting. See Plaintext Email and Comments on this site for more. Note also that comments are expected to be open, considerate, and respectful.
Comments (1)
2017-02-04 Ankit
I wish I found this sooner! This makes sooooo much sense.