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From: Cogito on 25 Apr 2010 17:29
Im relatively new to the "serious" evaluation techniques. As a
multivariate calc student, all I know thus far are the various
convergence and divergence tests, and of course the evaluation
equations for trivial series. But Im just curious as to how the "big
wigs" figure out what a series converges to... in the non-trivial
cases. Can someone run me through the highlights of the numerous
techniques that I can only imagine exist... give me terms - something
specific I can googlerize for myself at the very least. Its
impossible to find useful specifics when you dont have the
vocabulary... "evaluating a sum" doesnt produce very worthy results
for my needs.
From: fishfry on 25 Apr 2010 19:13
In article
<a990ea06-0b6b-4009-b5ce-529b8d608f52(a)h31g2000prl.googlegroups.com>,
Cogito <cogitoergocogitosum(a)gmail.com> wrote:

> Im relatively new to the "serious" evaluation techniques. As a
> multivariate calc student, all I know thus far are the various
> convergence and divergence tests, and of course the evaluation
> equations for trivial series. But Im just curious as to how the "big
> wigs" figure out what a series converges to... in the non-trivial
> cases. Can someone run me through the highlights of the numerous
> techniques that I can only imagine exist... give me terms - something
> specific I can googlerize for myself at the very least. Its
> impossible to find useful specifics when you dont have the
> vocabulary... "evaluating a sum" doesnt produce very worthy results
> for my needs.

You might be interested to look up the history of the Basel problem,
proposed in 1644 and solved by Euler in 1735. The problem is to find the
sum of the series 1 + 1/4 + 1/9 + 1/16 + ... + 1/n^2 + ..., which turns
out to be the suprising pi^2/6.

Euler's solution was hardly what we'd call rigorous today, but it
provides interesting insight into how a great mathematician attacks a
hard problem.

These days there are many rigorous proofs that you can study as a
learning exercise.
From: porky_pig_jr on 25 Apr 2010 20:36
On Apr 25, 5:29 pm, Cogito <cogitoergocogito...(a)gmail.com> wrote:
> Im relatively new to the "serious" evaluation techniques.  As a
> multivariate calc student, all I know thus far are the various
> convergence and divergence tests, and of course the evaluation
> equations for trivial series.  But Im just curious as to how the "big
> wigs" figure out what a series converges to... in the non-trivial
> cases.  Can someone run me through the highlights of the numerous
> techniques that I can only imagine exist... give me terms - something
> specific I can googlerize for myself at the very least.  Its
> impossible to find useful specifics when you dont have the
> vocabulary... "evaluating a sum" doesnt produce very worthy results
> for my needs.

To figure out whether the series converges at all is highly non-
trivial task. To figure out whether the convergent series converges to
something "known" (like in another reply, some function of pi) is even
more than non-trivial task and in fact this holds true for a very
small fraction of convergent series. Often we end up *defining* some
number (or function) as some convergent series. Like in analysis we
start with what you probably know as a Taylor Polynomials for sin and
cos functions. We write the series, prove that it converges and then
*define* both sin and cos as these series. So once again, just proving
the fact that the series converges is already non-trivial task. If you
are on a level of Calc III, probably you should take some introductory
book on Analysis (like Rudin, principles of analysis) and study the
chapters on sequences and series. The focus is on power series, and
how it interacts with Taylor expansion, so the rigorous treatment of
taylor theory is a must (also in Rudin). There are some techniques
available to determine whether the series converges, again in the same
textbook. A good example is proving the Stirling's formula. And not
that the ratio is sqrt(2 pi) but just that fact that it in fact
converges to some finite value. That's a hard one and it uses the
taylor expansion. probably a good example. Incidently, the series that
converges to pi^2/6 (in other reply) arise as a byproduct of Fourier
analysis.

From: amzoti on 25 Apr 2010 21:12
On Apr 25, 2:29 pm, Cogito <cogitoergocogito...(a)gmail.com> wrote:
> Im relatively new to the "serious" evaluation techniques.  As a
> multivariate calc student, all I know thus far are the various
> convergence and divergence tests, and of course the evaluation
> equations for trivial series.  But Im just curious as to how the "big
> wigs" figure out what a series converges to... in the non-trivial
> cases.  Can someone run me through the highlights of the numerous
> techniques that I can only imagine exist... give me terms - something
> specific I can googlerize for myself at the very least.  Its
> impossible to find useful specifics when you dont have the
> vocabulary... "evaluating a sum" doesnt produce very worthy results
> for my needs.

Check out the cheap Dover books:

http://store.doverpublications.com/0486601536.html

1. Konrad Knopp
2. James M Hyslop
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