CONTEXT
Common series are usually described as infinite sums, written as consecutive terms ending with (…). Or they can be described using the $\sum_{}$ symbol and arguments usually including $(-1)^k$ for alternating series or $(2k+1)$ for odd ranks series for instance.
Dirichlet character and moduli provide conciser descriptors for L-series.
Many non exhaustive grouped lists of series can be found.
QUESTION 1
On non-academic public access on-line sources, I could not find any table assembling and describing, even a limited number of, “usual” or “fundamental” series. Of course the point would not be exhaustivity, but it would rather be renewed perspective on and lighter writing of the description of series. To help get a better sense of some series patterns specificities.
So first, do you know of any such table ?
PROPOSITION
“Fundamental” can be subjective as a term, so this is a draft for 2 concise tables of “usual” series and of "usual" L-Series, the limits/sums of which describe squareroots, logarithm, exponential, several trigonometric functions or ratios and products of $\pi$.
I decided on a $\frac{1}{x}$ elementary argument to get a uniform description of limits. And the domain for now is limited to $/R$, but I am not sure it would be relevant to look for something else than a minimal common domain for all the functions thus described.
I opted for a description splitting series characteristics in two parts : patterns in columns (full integer series, alternate full integer series, odd series, alternate odd series, even series, alternate even) and arguments in rows.
For the second table, focusing on L-Functions, I also added two series patterns, which could be described with Dirichlet characters, but I use a simplified “n(2k+1)” as pattern name. It is describing series whose consecutive elements have a 2n period, a radius of 1 and a radius of (n-1). I described and illustrated these in a first question asked about the Arc to Chord ratio as an infinite sum.
(I am sorry this has to be pictures for now, I will try to mathjax the tables soon.)
(Arbitrary names (grayed) are given to some L-Series for easier use as argument)
QUESTION 2
What would you recommend adding to these tables ? I’m considering adding a row for the $\frac{1}{n}$ argument in the first table. Many functions also have an exponential or logarithmic expression which could constitute another full specific table. But some functions only have exponential or logarithmic expression. I deliberately described series including an L-function in their argument, as even and alternate even patterns, considering the L-functions structures it sound relevant that way, but they could be described as full and alternate full too.
