Talk:Convergence

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Contents

Checklist for Writing Pages

Added by Greg Brown.

Messages to the Future

  • The page is long already. I don't think much more would need to be added.
  • Potentially the page could be adapted for complex numbers in sequences and series, but I doubt another page would need such information.
  • The Geometric Series page could probably be developed a bit. That page deals with mostly different topics, but it could be cleaned up. The pages would make good "companions."
  • The page could probably use some more images. Will mention below.

Reference and Footnotes

  • No in-text citations needed.
  • All images taken from elsewhere are cited.

Good Writing

Context

  • It's a helper page, so the focus is not so much on why it's interesting, hook, etc.
  • It does synchronize well with the Taylor Series page: the power series info refers to Taylor series, and the natural log Taylor series is used to determine the sum of the alternating harmonic series.

Quality of Prose and Page Structure

  • The sequence of sections makes sense. It defines the necessary terms/notation and then goes on to show what convergence means in general. Then it shows how to determine it using tests. Then it gives the examples of (alternating) harmonic series.
  • I state clearly what each section is for. I also define important terms clearly and at (I think) appropriate times.
  • Examples and proofs in hidden sections. The page is structured so that it would be useful as a resource, ie. you can skip to a section you want, read what the theorem is, look at some examples of how it's used if you want, or see how it is proved. The proofs are not entirely necessary to understanding how to use the tests or (in some cases) even how they work, as some of them are pretty intuitive - so that is why the proofs come after the examples.

Integration of Images and Text

  • Not a ton of images, because it's a helper page and they are not always necessary.
  • Tried to add some images to the definition of convergence section, as well as a few of the tests, namely Comparison and Integral Tests. There could definitely be an image for Alternating Series Test, but I did not get around to finding/making one.

Connection to Other Mathematical Topics

  • Mostly references to Taylor series.
  • There are also references to geometric series. The Geometric Series page probably needs to be cleaned up a bit to be a sufficient reference, though.
  • There is also a reference to Riemann rectangles for the Integral Test.

Examples, Calculations, Applications, Proofs

  • Lots of examples and proofs.
  • Some explicit calculations, particularly it is possible to sum series.

Mathematical Accuracy and Precision of Language

  • I've tried to make the math language very precise.
  • Notation set up early and used consistently throughout.
  • Proofs are well-stated.
  • Not much "strange" notation. The page is fairly accessible, I think.

Layout

  • Some of the images are large, width-wise. But they should not cause issues for smaller window sizes.
  • Extensive hiding of proofs and examples.
  • All terms are bold-faced when they are defined.

General Feedback

Notation for Sequences and Series

Convergence and Divergence

In general, we don’t talk about n “becoming” arbitrarily large – rather, we consider an arbitrarily large value of n. So you might say, “an approaches a single number for arbitrarily large n.” I don’t think what you have here is technically inaccurate, but it’s sort of misleading. It's very common for students to misunderstand a limit as a dynamic object (because of words like "approaches"), so it's good to avoid language that could imply that.

The use of the -∞ < L < ∞, while pretty common, is a bit confusing, especially when you haven't specified what number system you're in. I realize most people will assume we're talking about real or real and complex numbers, but I might actually say Ln ∈ ℝ, ℂ. That might be too constraining; I don't know for how many spaces this definition holds. But saying a value is between negative and positive infinity without saying which system you're in is technically pretty vague. (Though maybe people will make the appropriate assumption so naturally that it doesn't matter?)

After your first expression for the limit of a series, provide the same in summation notation, where:

\lim_{n\rightarrow\infty}\sum^n_{k=1}a_n=L_s

... which also leads well into the partial-sum expression.

In the first example in this section, you give the first five partial sums of a series, then say, "In general, then:" and give the formula for sn. However, that pattern won't be obvious to many readers; either omit the word "then," or actually show that claim to be true. (When you do the same in the second example, it is obvious enough to do it that way.)

-Diana 14:14, 17 July 2013 (EDT)

Convergence Tests

nth-term Test

[Note: This is a point that we discussed in a meeting. at the time of this post, this section has been edited.] In your explanation, you say:

Imagine that a sequence converges to some L_A \neq 0. Then the series s is the sum of an infinite number of terms, all of which are not 0. If s_n = L_s, then for all n, it will be true that s_{n+1} = L_s + a_{n+1} \neq L_s. The series cannot converge.

First, note that a series can converge to a non-zero value and still have terms in it that are zero. Second, most series are made up of non-zero values, in which case s_{n+1} = L_s + a_{n+1} \neq L_s does indeed hold for any specific value of n. What is important is that, if the limit of the related sequence is non-zero, then the limit of the difference between consecutive partial sums cannot be zero. That is, let the sequence have non-zero limit a, and assume the partial sum for arbitrarily large n is Ln. Then the partial sum at n + 1 is Ln + a. So \lim_{n\rightarrow\infty}L_{n+1}-L_n=a. That may seem trivial, but it necessarily contradicts convergence of the series by a definition you haven't included here: An infinite series converges if and only if for every ε > 0 there exists a value N such that for m > n > N, |an + ... + am| < ε.

-Diana 14:14, 17 July 2013 (EDT)

Comparison Test

This section would benefit from images. I tend to picture the comparison test sort of as the “squeeze theorem” is often illustrated; the series we are using for the test operates as a bound for the series we are testing, and you can show this graphically. Such an illustration would also help show why beginning the test at some N is valid.

Be careful how you use infinity; it cannot be a sum in the number systems we're using here.

-Diana 14:14, 17 July 2013 (EDT)

Limit Test

Ratio Test

Root Test

Integral Test

Power Series

[Note: This is a point that we discussed in a meeting. at the time of this post, this section has been edited.] You say that, "It is also possible for a series to be absolutely convergent; that is, for the series to converge regardless of the value of the variable," but this is not exactly correct. A series s = Σ an is absolutely convergent if t = Σ |an| is convergent. Also, absolute convergence is something you should mention in your convergence tests, since it means the ones that nominally only apply to positive series can be adjusted to work with series with negative values, too.

-Diana 14:14, 17 July 2013 (EDT)

Harmonic Series

Alternating Harmonic Series

Examples of Series

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