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I've read that there are methods called series accelerations or sequence transformations that can help you make some divergent series converge and convergent series converge over a longer interval. I have a book devoted to that specific topic, but I can't understand it too well (It is written for professional research mathematicians and probably doesn't contain a single concrete example in the whole book). I want to know what you would recommend that teaches about series accelerations, one's that can make power series converge over longer intervals, at the undergraduate level preferably. Well just English (as opposed to symbolic) explanations and some examples. Or you can just tell me what books you found it easy to learn from. I really don't even know where most people learn about it. Thank you.

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It'd help a lot if you said which book you have. –  Zev Chonoles Mar 26 '12 at 4:51
It's called Practical Extrapolation Methods: Theory and Applications by Avram Sidi. –  Kenny Mar 26 '12 at 5:17
I don't think there's any such thing as a series transformation that makes a convergent series converge over a longer interval. Also, what series acceleration does is it makes a convergent series converge faster. Books with titles like "Numerical Methods" or "Numerical Analysis" will often have a chapter or two on these techniques. –  Gerry Myerson Mar 26 '12 at 5:19
You're probably right. I probably just thought I read it in there somewhere, but I just now looked over the paragraphs where I thought it should be and didn't find it. I think it can make some divergent series convergent though. Don't take my word on that. Anyways I am only interested in making the interval of convergence longer. If I can't do that I need to solve my problems some other way. I'll ask another question about it sometime. –  Kenny Mar 26 '12 at 5:44
Perhaps, Kenny, you are thinking of a technique called analytic continuation. E.g., the series $1+z+z^2+\dots$ converges only for $|z|\le1$, but it can be analytically continued to the function $f(z)=(1-z)^{-1}$, which is defined everywhere except at $z=1$. Then you can find series for $f$ which converge at places where the original series didn't. This is usually studied in courses on complex analysis (funcitons of a complex variable). –  Gerry Myerson Mar 26 '12 at 23:31

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The way I see it, there really isn't a book that caters to undergraduates that deals properly with sequence transformations. It is my opinion that sequence transformations are a bit like power tools: they can lead to disaster if used improperly. (There is the cautionary tale that quite a number of these sequence transformations yield something close to 0.5 as the result if fed the output of a uniform pseudorandom number generator.) The standard books, Brezinski and Redivo-Zaglia's Extrapolation Methods: Theory and Practice and Sidi's book which you have already mentioned are a tad too advanced for undergraduate perusal. Weniger's long survey article might be a bit more useful to you, as well as Appendix A of The SIAM 100-Digit Challenge: A Study in High-Accuracy Numerical Computing by Dirk Laurie, as they are slightly more application-oriented than theory oriented (but again, I must stress that it's way too easy to get into trouble if you do not understand the theory at all).

One of the more spectacular examples of making series converge in regions past their usual domain of convergence is provided by the (violently!) divergent alternating series

$$\frac{\exp(1/z)}{z}E_1(1/z)\sim \sum_{k=0}^\infty k! (-z)^k$$

If you try generating the partial sums for $z=2$, you find a seemingly messy sequence that just keeps increasing in magnitude. However, if one applies the Shanks transformation to the violently divergent sequence, one obtains the expected answer; we see here that a power series with zero radius of convergence can be made to yield "correct" answers through some massaging.

Here is a Mathematica demonstration. High precision is very much needed here since there will be a fair bit of subtractive cancellation during the numerical evaluation:

seq = N[Accumulate[Table[k! (-2)^k, {k, 0, 20}]], 50];

SequenceLimit[seq, Method -> {"WynnEpsilon", Degree -> 1}]

% - (Exp[1/2] ExpIntegralE[1, 1/2]/2)

Admittedly the number of "good" digits isn't that much, but one should still be surprised that a usable result was obtainable at all...

(See also this previous answer I gave.)

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