ch-ch-update: series, closed-form series, and strides

[Lars Kyllingstad]

Andrei Alexandrescu wrote:
> Great! I didn't know (haven't learned college-level Math in English; 
> sometimes I wonder how I fumbled through grad school without major 
> misunderstandings). By the way, I might have been wrong with the name 
> "series" itself. I thought "series" is something like a_n = 
> f(a_{n-1},...,f_a{n-k}). However, according to Wikipedia:
> 
> http://en.wikipedia.org/wiki/Infinite_series
> 
> series is really what I thought is called "partial sums", i.e. s_n is 
> the sum of elements of a sequence a_n up to the nth element.
> 
> So should I change "series" with "sequence"? How about what I called 
> "ClosedFormSeries"? By that I meant a series, (pardon, sequence), in 
> which there is no recurrence formula - the nth element a_n can be 
> expressed in terms of n and a[0], ..., a[k] (a sort of "random access" 
> for a sequence).
> 
> So, what names should I use? English-speaking mathematicians across the 
> newsgroup, unite!

I believe the mathematically correct terms would be RecursiveSequence 
and ClosedFormSequence. If I were to use just Sequence, it would in fact 
be for the latter.

A series is, as you said, the sum of all the terms in a sequence, 
whereas summing only a finite set of terms gives a partial sum. (The 
latter could be implemented as a range.)

A recurrence relation is an expression that recursively defines a 
sequence (i.e. the string argument of the template).

I prefer MathWorld to Wikipedia when it comes to these things. Here are 
some (possibly) useful links:

Recursive sequences:
     http://mathworld.wolfram.com/RecursiveSequence.html

Some sequences that could possibly be rangeified:
     http://mathworld.wolfram.com/Sequence.html

Series:
     http://mathworld.wolfram.com/Series.html

Partial sums (and other sums) of sequences:
     http://mathworld.wolfram.com/PartialSum.html


-Lars