Re: inverse command of Series[]

*To*: mathgroup at smc.vnet.net*Subject*: [mg8067] Re: inverse command of Series[]*From*: Paul Abbott <paul at physics.uwa.edu.au>*Date*: Mon, 4 Aug 1997 01:47:50 -0400*Organization*: University of Western Australia*Sender*: owner-wri-mathgroup at wolfram.com

Nguyen N. Anh wrote: > I've written a program in Mathematica and it works quite well. However, after > convergence it gave as result the following polinomial: > > 1 + x + x^2/2 + x^3/6 + x^4/24 + x^5/120 + x^6/720 + x^7/5040 > > I know that this is the power series expansion of function E^x but I want that > my program recognizes it automatically and outputs the generating function of > the polinomial calculated ( E^x in this case). > > Is there any command or package which can do that ? Have a look at the Mathematica Journal 5(3): 33-35. I have included part of that discussion below: Suppose that you have the first few terms of a series, say: series = 1 - 2 x + 3 x^2 - 4 x^3 + 5 x^4; Often you would like to obtain the n-th term of the series and a closed form expression, if possible. In general this is impossible as there are an infinite number of functions that have identical series expansions up to a given degree. However, if extra information is available (such as the recurrence relation), one general method for compacting such expressions is to use generating functions. After loading << DiscreteMath`RSolve` and providing the recurrence relation for the series: RSolve[{a[n+1] == -a[n] - (-1)^n, a[0] == 1}, a[n], n] n {{a[n] -> (-1) (1 + n)}} one finds that the n-th term is a[n] x^n /. First[%] n n (-1) (1 + n) x and a closed form for the infinite series can be obtained using PowerSum: PowerSum[a[n] /. First[%%], {x, n, 0}] // Simplify -2 (1 + x) As a check, we expand this out into a Maclaurin series: % + O[x]^5 2 3 4 5 1 - 2 x + 3 x - 4 x + 5 x + O[x] An alternative approach is to note that a large class of functions have hypergeometric series expansions. In the Advanced Tutorial notes for the 1992 Mathematica Conferences, Kelly Roach outlines a simple algorithm for converting Maclaurin series into hypergeometric functions. Given a series expansion then, if the ratios yn/(yn+1) are simple rational functions in n, then we can express the series as a hypergeometric function. With some work this could be turned into a package for converting the first few terms of a hypergeometric series into the corresponding (minimal) hypergeometric function. Nevertheless, it is better to work through the problem step-by-step, checking a large number of cases instead of trying to write such a package from scratch. Cheers, Paul ____________________________________________________________________ Paul Abbott Phone: +61-8-9380-2734 Department of Physics Fax: +61-8-9380-1014 The University of Western Australia Nedlands WA 6907 mailto:paul at physics.uwa.edu.au AUSTRALIA http://www.pd.uwa.edu.au/Paul God IS a weakly left-handed dice player ____________________________________________________________________

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**Re: inverse command of Series[]**