differentialorder: a positive integer n (default: n=4); upper bound for the order of the differential equation searched for. This controls the depth of the search for a differential equation for expr. Higher values of n will increase the chance to find the solution, but increase the running time as well.

dir: one of default, left, right, real, or complex; direction of the limit computation for initial values. If a is finite, then the default is dir=complex. If a is either infinity or -infinity, then the default is dir=real. See also limit.

makereal: either true or false (default); makereal=true (or just makereal for short) indicates that a series with real coefficients should be returned

method: one of default, hypergeometric, rational, or exponential. Specifies the method that will be used; the default method uses an internal selection strategy. See the Examples below for an illustration of the various methods.

recurrence: either true or false (default). If recurrence=true (or recurrence for short) is given and no formal power series can be computed, then the output is a recurrence for $b\left(k\right)$.

This command expands meromorphic functions of certain type into their corresponding Laurent-Puiseux series as a sum of terms of the form $\sum _{k=0}^{\mathrm{\infty }}b\left(k\right){\left(x-a\right)}^{\frac{mk+s}{q}}$, where m is called the symmetry number, s is the shift number, and a is the expansion point.

The following types are supported:

functions of hypergeometric type, where $\frac{b\left(k+m\right)}{b\left(k\right)}$ is a rational function of k for some integer m;

functions of exponential type, which satisfy a linear homogeneous differential equation with constant coefficients;

functions of rational type, which are either rational or have a rational derivative;  

linear combinations of hypergeometric functions are treated by the Petkovsek-van-Hoeij algorithm; see LREtools[hypergeomsols].

The convert(expr, FormalPowerSeries, x=a) command tries to find a formal power series expansion for expr with respect to the variable x at the point of expansion a. If a=infinity, then the command searches for an asymptotic series. It also works for formal Laurent-Puiseux series, and in certain cases of logarithmic singularities.

The command first looks for a homogeneous linear differential equation with polynomial coefficients for expr; hence Maple must know the derivatives of expr.

If eq is a variable name x, or if eq is omitted and expr has only one variable x, then x=0 is assumed.

The optional argument k is a name that will be taken as the summation variable in the result. If it is not specified, then one of the variable names k, k0, k1, etc. is chosen.

To compute asymptotic power series, one may expand the function around $\mathrm{\infty }$; see the Examples below. The result is a (possibly divergent) series.

The FormalPowerSeries argument can be abbreviated as FPS.

For a complete list of known functions, see inifcns.

The convert/sum command provides the same functionality as the convert/FormalPowerSeries command, with a newer algorithm which can employ alternate methods.

$\mathrm{convert}\left(\sin \left(x\right)\,\mathrm{FormalPowerSeries}\right)$

$\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(\mathrm{-1}\right)}^k{x}^{2k+1}}{\left(2k+1\right)!}$

$\mathrm{convert}\left(\ln \left(x\right)\,\mathrm{FormalPowerSeries}\,x=1\,j\right)$

$\sum _{j=0}^{\mathrm{\infty }}\frac{{\left(\mathrm{-1}\right)}^j{\left(x-1\right)}^{j+1}}{j+1}$

$\mathrm{convert}\left(\mathrm{sqrt}\left(\frac{1-\mathrm{sqrt}\left(1-x\right)}{x}\right)\,\mathrm{FormalPowerSeries}\,x\,n\right)$

$\sum _{n=0}^{\mathrm{\infty }}\frac{2^{-4n+\frac{1}{2}}\left(4n\right)!x^n}{2{\left(2n\right)!}^2\left(2n+1\right)}$

$\mathrm{convert}\left(\exp \left(\arcsin \left(x\right)\right)\,\mathrm{FormalPowerSeries}\right)$

$\left(\sum _{k=0}^{\mathrm{\infty }}\frac{\left(\prod _{j=0}^k\left(4j^2+1\right)\right)x^{2k}}{\left(4k^2+1\right)\left(2k\right)!}\right)+\left(\sum _{k=0}^{\mathrm{\infty }}\frac{2^k\left(\prod _{j=0}^k\left(2j^2+2j+1\right)\right)x^{2k+1}}{\left(2k^2+2k+1\right)\left(2k+1\right)!}\right)$

$\mathrm{convert}\left(\frac{t}{1-xt-t^2}\,\mathrm{FormalPowerSeries}\,t\right)$

$\sum _{k=0}^{\mathrm{\infty }}\left(-\frac{{\left(\frac{x}{2}-\frac{\sqrt{x^2+4}}{2}\right)}^k}{\sqrt{x^2+4}}+\frac{{\left(\frac{x}{2}+\frac{\sqrt{x^2+4}}{2}\right)}^k}{\sqrt{x^2+4}}\right)t^k$

$\mathrm{convert}\left(\ln \left(\frac{1+x^2}{1-x}\right)\,\mathrm{FormalPowerSeries}\,\mathrm{method}=\mathrm{rational}\right)$

$\sum _{k=0}^{\mathrm{\infty }}\frac{\left({\left(\mathrm{-I}\right)}^k{\mathrm{I}}^k+\mathrm{I}{\left(\mathrm{-I}\right)}^k-\mathrm{I}{\mathrm{I}}^k\right)x^{k+1}}{{\left(\mathrm{-I}\right)}^k{\mathrm{I}}^k\left(k+1\right)}$

$\mathrm{convert}\left(\exp \left(x\right)\sin \left(x\right)\,\mathrm{FormalPowerSeries}\,\mathrm{method}=\mathrm{hypergeometric}\right)$

$\sum _{k=0}^{\mathrm{\infty }}\left(\frac{\mathrm{I}{\left(1-\mathrm{I}\right)}^k}{2k!}-\frac{\mathrm{I}{\left(1+\mathrm{I}\right)}^k}{2k!}\right)x^k$

$\mathrm{convert}\left(\exp \left(x\right)-2\exp \left(-\frac{x}{2}\right)\cos \left(\frac{\mathrm{sqrt}\left(3\right)x}{2}+\frac{\mathrm{\pi }}{3}\right)\,\mathrm{FormalPowerSeries}\,\mathrm{method}=\mathrm{exponential}\right)$

$\sum _{k=0}^{\mathrm{\infty }}\left(\frac{1}{k!}-\frac{\cos \left(\frac{2k\mathrm{\pi }}{3}\right)}{k!}+\frac{\sqrt{3}\sin \left(\frac{2k\mathrm{\pi }}{3}\right)}{k!}\right)x^k$

$\mathrm{convert}\left(\exp \left(x\right)-2\exp \left(-\frac{x}{2}\right)\cos \left(\frac{\mathrm{sqrt}\left(3\right)x}{2}+\frac{\mathrm{\pi }}{3}\right)\,\mathrm{FormalPowerSeries}\,\mathrm{method}=\mathrm{hypergeometric}\right)$

$\sum _{k=0}^{\mathrm{\infty }}\frac{3x^{3k+1}}{\left(3k+1\right)!}$

$\mathrm{convert}\left(\exp \left(\mathrm{sqrt}\left(x\right)\right)\,\mathrm{FormalPowerSeries}\right)$

$\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(\sqrt{x}\right)}^k}{k!}$

$\mathrm{convert}\left(\frac{\sin \left(x\right)}{x^6}\,\mathrm{FormalPowerSeries}\right)$

$\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(\mathrm{-1}\right)}^k{x}^{2k-5}}{\left(2k+1\right)!}$

`diff/g` := proc(a,x) g(a)*diff(a,x) end proc:

$\mathrm{convert}\left(g\left(x\right)\,\mathrm{FormalPowerSeries}\,x=0\right)$

$\sum _{k=0}^{\mathrm{\infty }}\frac{g\left(0\right)x^k}{k!}$

$\mathrm{convert}\left(\mathrm{Int}\left(\frac{\mathrm{erf}\left(t\right)}{t}\,t=0..x\right)\,\mathrm{FormalPowerSeries}\,x\right)$

$\sum _{k=0}^{\mathrm{\infty }}\frac{2{\left(\mathrm{-1}\right)}^k{x}^{2k+1}}{\sqrt{\mathrm{\pi }}k!{\left(2k+1\right)}^2}$

$\mathrm{convert}\left(\left(-\frac{1}{2}x+\frac{1}{6}{x}^3\right)\arctan \left(x\right)+\left(-\frac{1}{4}{x}^2+\frac{1}{12}\right)\ln \left(x^2+1\right)+\frac{5}{12}{x}^2+\frac{1}{4}\,\mathrm{FormalPowerSeries}\,x=0\,\mathrm{makereal}\right)$

$\frac{1}{4}+\left(\sum _{k=0}^{\mathrm{\infty }}\frac{x^{k+4}\cos \left(\frac{k\mathrm{\pi }}{2}\right)}{\left(k+4\right)\left(k+3\right)\left(k+2\right)\left(k+1\right)}\right)$

$\mathrm{convert}\left(\sin \left(x+y\right)\,\mathrm{FormalPowerSeries}\,x\right)$

$\left(\sum _{k=0}^{\mathrm{\infty }}\frac{\sin \left(y\right){\left(\mathrm{-1}\right)}^k{x}^{2k}}{\left(2k\right)!}\right)+\left(\sum _{k=0}^{\mathrm{\infty }}\frac{\cos \left(y\right){\left(\mathrm{-1}\right)}^k{x}^{2k+1}}{\left(2k+1\right)!}\right)$

$\mathrm{convert}\left(\frac{1}{x^4+x+1}\,\mathrm{FormalPowerSeries}\,\mathrm{method}=\mathrm{rational}\right)$

$\sum _{k=0}^{\mathrm{\infty }}\left(\sum _{\mathrm{\_\α}=\mathrm{RootOf}\left({\mathrm{\_Z}}^4+\mathrm{\_Z}+1\right)}\frac{36{\mathrm{\_\α}}^3-48{\mathrm{\_\α}}^2+64\mathrm{\_\α}+27}{229{\mathrm{\_\α}}^{k+1}}\right)x^k$

$\mathrm{convert}\left(\mathrm{AiryAi}\left(x\right)\,\mathrm{FormalPowerSeries}\right)$

$\left(\sum _{k=0}^{\mathrm{\infty }}\frac{3^{-2k+\frac{1}{3}}{x}^{3k}}{3\mathrm{\Gamma }\left(\frac{2}{3}\right)\mathrm{pochhammer}\left(\frac{2}{3}\,k\right)k!}\right)+\sum _{k=0}^{\mathrm{\infty }}\left(-\frac{\mathrm{\Gamma }\left(\frac{2}{3}\right)3^{-2k+\frac{1}{6}}{x}^{3k+1}}{2\mathrm{\pi }\mathrm{pochhammer}\left(\frac{4}{3}\,k\right)k!}\right)$

$\mathrm{convert}\left(\mathrm{hypergeom}\left(\left[a\,b\right]\,\left[c\right]\,x\right)\,\mathrm{FormalPowerSeries}\,x\right)$

$\sum _{k=0}^{\mathrm{\infty }}\frac{\mathrm{pochhammer}\left(b\,k\right)\mathrm{pochhammer}\left(a\,k\right)x^k}{\mathrm{pochhammer}\left(c\,k\right)k!}$

$\mathrm{convert}\left(\cos \left(4\arccos \left(x\right)\right)\,\mathrm{FormalPowerSeries}\right)$

$8x^4-8x^2+1$

$f≔\mathrm{expand}\left({\left({\cos \left(x\right)}^2+{\sin \left(x\right)}^2\right)}^{10}\right)$

$f≔{\cos \left(x\right)}^{20}+10{\cos \left(x\right)}^{18}{\sin \left(x\right)}^2+45{\cos \left(x\right)}^{16}{\sin \left(x\right)}^4+120{\cos \left(x\right)}^{14}{\sin \left(x\right)}^6+210{\cos \left(x\right)}^{12}{\sin \left(x\right)}^8+252{\cos \left(x\right)}^{10}{\sin \left(x\right)}^{10}+210{\cos \left(x\right)}^8{\sin \left(x\right)}^{12}+120{\cos \left(x\right)}^6{\sin \left(x\right)}^{14}+45{\cos \left(x\right)}^4{\sin \left(x\right)}^{16}+10{\cos \left(x\right)}^2{\sin \left(x\right)}^{18}+{\sin \left(x\right)}^{20}$

$\mathrm{convert}\left(f\,\mathrm{FormalPowerSeries}\right)$

$1$

$\mathrm{convert}\left(\exp \left(x^2\right)\left(1-\mathrm{erf}\left(x\right)\right)\,\mathrm{FormalPowerSeries}\,x=\mathrm{\infty }\,\mathrm{dir}=\mathrm{left}\right)$

$\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(\mathrm{-1}\right)}^k\left(2k\right)!4^{-k}{\left(\frac{1}{x}\right)}^{2k+1}}{\sqrt{\mathrm{\pi }}k!}$

$\mathrm{convert}\left(\sin \left(\frac{1}{x}\right)\,\mathrm{FormalPowerSeries}\,x=\mathrm{\infty }\right)$

$\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(\mathrm{-1}\right)}^k{\left(\frac{1}{x}\right)}^{2k+1}}{\left(2k+1\right)!}$

$\mathrm{convert}\left(\exp \left(-\frac{1}{x^2}\right)\,\mathrm{FormalPowerSeries}\,x=0\,\mathrm{dir}=\mathrm{real}\right)$

$0$

$\mathrm{convert}\left(\exp \left(-\frac{1}{x}\right)\,\mathrm{FormalPowerSeries}\,x=0\,\mathrm{dir}=\mathrm{right}\right)$

$0$

$\mathrm{convert}\left(\exp \left(-\frac{1}{x}\right)\,\mathrm{FormalPowerSeries}\,x=0\,\mathrm{dir}=\mathrm{left}\right)$

${\ⅇ}^{-\frac{1}{x}}$

$\mathrm{convert}\left(\exp \left(\frac{1}{x}\right)\,\mathrm{FormalPowerSeries}\right)$

${\ⅇ}^{\frac{1}{x}}$

$\mathrm{convert}\left(\tan \left(x\right)\,\mathrm{FormalPowerSeries}\right)$

$\tan \left(x\right)$

$\mathrm{convert}\left({\arcsin \left(x\right)}^3\,\mathrm{FormalPowerSeries}\right)$

${\arcsin \left(x\right)}^3$

$\mathrm{convert}\left({\arcsin \left(x\right)}^3\,\mathrm{FormalPowerSeries}\,b\left(k\right)\,\mathrm{recurrence}\right)$

$k^{4}b\left(k\right)-2\left(k+1\right)\left(k+2\right)\left(k^2+2k+2\right)b\left(k+2\right)+\left(k+1\right)\left(k+2\right)\left(k+3\right)\left(k+4\right)b\left(k+4\right)=0$

$\mathrm{convert}\left(\mathrm{arcsech}\left(x\right)\,\mathrm{FormalPowerSeries}\right)$

$\ln \left(2\right)+\ln \left(\frac{1}{x}\right)+\sum _{k=0}^{\mathrm{\infty }}\left(-\frac{\left(2k+1\right)!4^{-k}{x}^{2k+2}}{2{k!}^2\left(k+1\right)\left(2k+2\right)}\right)$

Gruntz, Dominik, and Koepf, Wolfram. "Maple package of formal power series."  Maple Technical Newsletter, Vol. 2(2), (1995):22-28.

Koepf, Wolfram. "Algorithmic development of power series. Artificial intelligence and symbolic mathematical computing." Lecture Notes in Computer Science, Vol. 737, pp. 195-213. Edited by J. Calmet and J. A. Campbell.  Berlin-Heidelberg: Springer, 1993.

Koepf, Wolfram. "Examples for the algorithmic calculation of formal Puiseux, Laurent and power series." SIGSAM Bulletin Vol. 27, (1993): 20-32.

Koepf, Wolfram. "Power series, Bieberbach conjecture and the de Branges and Weinstein functions." Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation. pp. 169-175. New York: ACM, 2003.

Koepf, Wolfram. "Power series in computer algebra." Journal of Symbolic Computation, Vol. 13, (1992): 581-603.

van Hoeij, Mark. "Finite singularities and hypergeometric solutions of linear recurrence equations." J. Pure and Appl. Algebra, Vol. 139, (1999): 109-131.