What is a good, easy to read book that tells you how to make series converge over a longer interval?

Question

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.

Answer 1

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}]
0.4614551195483545958`6.020599913279623

% - (Exp[1/2] ExpIntegralE[1, 1/2]/2)
0``6.356470444711438

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.)