(July 21, 2023 at 2:54 pm)Loaded dice Wrote:(March 8, 2023 at 4:47 pm)GrandizerII Wrote: Doing algebra with infinite series is tricky
It's wrong. All the basic rules you know about manipulating and rearranging the terms of a finite sum, are no longer true in general when it comes to infinite series.
The first thing to do with an infinite series is to study its convergence. If it is convergent, in some cases it's possible to rearrange its terms without losing convergence (e.g. https://en.wikipedia.org/wiki/Riemann_series_theorem)
It's useful to remember that the entire topic of infinite series is serious undergraduate math, if you're not familiar with basic real analysis and sequences just forget them for now.
I'm just laughing at the phrase 'serious undergraduate math'. Infinite series are usually done either at the end of calculus 2 or in calculus 3. As such, they are a topic for freshmen and sophomores. And such do not have to be math majors.
A decent advanced calculus class will prove and extend the topics seen in calculus, perhaps even addressing issues of uniform convergence of series of functions (as opposed to simply absolute or conditional convergence of numerical series) and contrasting such with pointwise convergence.
And, like I noted before, there are extended notions of convergence for series. Look up Cesaro and Abel summability, for example. Or, if you really feel adventuresome, try a more general Tauberian result. If you want references, I can provide them.
