Suppose one has found the complete Taylor series of a certain function during research. Afterwards, one would like to find the corresponding closed-form expression of this function -- assuming it is analytic. What are the different ways to go about that process? I know there is a bag of tricks, including multiplying the Taylor series with $x$ or differentiating it to try to find a Taylor series that is already-well-known. But these seem like ad-hoc approaches to me. So my questions are:
- Is there a systematic approach to find the closed-form expression of some Taylor Series for which the general term is known?
- Suppose one has found a closed-form expression for which the Taylor Series matches the series you have found for the first $k$ terms, but afterwards they differ. Does a method exist with which one can adjust the closed form in such a way that its Taylor series will coincide with the series you've found after the $k$'th term as well? (So it does necessarily not have to wholly coincide with the series one is after, but at least one get get successively closer to it by means of some iterative method of adjustment of the closed-form)
- Is there an overview article of the aforementioned “bag of tricks” to retrieve the closed-form by means of educated guesses to adjust the Taylor Series to make it look like a series that is well-known?
Let $n$ be the number of summands. If you got a closed-form expression for the finite summation problem ($n\in\mathbb{N}$), you could try to calculate the limit of that expression for $n\to\infty$.
Look for "Hypergeometric Summation", "Summation in finite terms" and "Symbolic summation".
There are i. a. decision algorithms for that.
Read e.g. the chapter "Symbolic Summation" in Bona, Miklos: Handbook of Enumerative Combinatorics. Chapman and Hall/CRC 2015.
There is a theory or an algorithm from Michael Karr:
Karr, Michael: Summation in finite terms. J. Assoc. Comp. Mach. 28 (1981) (2)305-350
Karr, Michael: Theory of Summation in Finite Terms. J. Symbolic Computation 1 (1985) (3) 303-315
And there is a theory or an algorithm from Carsten Schneider:
Look for
Schneider Summation
and for
Schneider sums