What branch of analysis deals most with sequences and series?

Question

I'm really interested in sequences and series (mainly series). What kind of math branch should I look more into? I understand that sequences and series mostly point toward analysis, but what sub-branch of analysis would I most enjoy?

For the sake of context, I am a senior in high school and I've taken single variable and multivariable calculus, linear algebra, and elementary differential equations.

I'm currently taking an introductory course on real analysis at a local university.

Thanks in advance for all the help!

Answer 1

It's not entirely true that sequences and series are the province of analysis. Generating functions take the form of discrete infinite sums and series and they are important tools in combinatorics and algebra. That being said, the study of sequences and series are mostly in analysis.

You should be warned,though,that as tools in real analysis and calculus, infinite series have largely been supplanted by direct numerical approximation methods.I'm extremely troubled by this drift because I think they are extraordinarily powerful tools in analysis-expressing functions locally as power series in a radius of convergence is very illuminating in "modeling" a function's behavior.But mathematicians-particularly numerical analysts-see infinite series as archaic tools for solving differential equations and look to rely on more rapidly convergent methods of approximation.

In complex analysis, however, where functions have the much stronger conditions of holomorphicity, infinite series and sequences still play an enormously important role to not only quantify analyticity, but to define the properties of contour integrals where singularities exist along paths. The main tool for this is Laurent series. I think you'll find a great deal of fascinating material to mine in complex analysis along these lines, usually presented in graduate complex analysis.

Answer 2

Wilf's book generatingfunctionolgy, available freely online, leads off with a quick list of the ways infinite series are used in combinatorics (where they are called generating functions); he immediately follows this up with some nice examples. The first time one encounters this sort of thing, it seems like magic. Glance through the table of contents and the first chapter to get the flavor.

The free online article by Carl Offner "A Little Harmonic Analysis" gives a nice introduction to the theory of Fourier series. To quote from his webpage:

As I was writing this up, I got interested in some historical questions. At the end of the paper I include a historical sketch that includes my views on two controversial topics:

• Did Abel prove "Abel's theorem" on the convergence of power series? (Yes, he did.)
• Did Dirichlet really come up with the modern definition of function? (I think it's quite reasonable to say that he did.)

and also my thoughts on a question that I have not seen dealt with seriously before:

• Why was Fejér's theorem such a sensation, since the essential results had been known for many years?

You might enjoy learning about $p$-adics, important in number theory, where the rationals are completed using a different metric than the usual one; in the 2-adics, for example, $1+2+4+\ldots 2^n+\ldots$ converges just fine.

Also a little off the beaten track (at the undergrad level, at least), the theory of divergent series contains many gems; the Wikipedia Article provides a nice starting point.

Finally, let me echo another poster: series really shine in complex analysis. The sooner any student encounters this branch of math, the better. In 300 years it has lost none of its lustre.