The series function computes a truncated series expansion of expr, with respect to the variable x, about the point a, up to order n. If a is infinity then an asymptotic expansion is given.

If eqn evaluates to a name x then the equation $x=0$ is assumed.

If the third argument n is present then it specifies the "truncation order" of the series calculations. This does not mean the truncation order of the actual series.  See Order for more information about this. If n is not present, the "truncation order" is determined by the global variable Order. The user may assign any non-negative integer to Order.  The default value of Order is 6.  See Order for more information.

If the series is not exact then an "order term" (for example, $\mathrm{O}\left(x^6\right)$ ) is the last term in the series.

It is possible to invoke series on user-defined functions. For example, if the procedure `series/f` is defined then the function call series(f(x,y),x)  will invoke `series/f`(x,y,x)  to compute the series. Note that this user-defined function `series/f` must return a series data structure, not just a polynomial (see type/series).

If series is applied to an unevaluated integral then the series expansion of the integral will be computed (if possible).

The result of the series function is a generalized series expansion. This could be a Taylor series or a Laurent series or a more general series. Formally, the coefficients in a "generalized series" are such that

for some constants $k_1$ and $k_2$, for any $0<\mathrm{ϵ}$ as $x$ approaches $a$. In other words, the coefficients may depend on x but their growth must be less than the polynomial in x. The order term may also hide such a coefficient, rather than an arbitrary constant. E.g., series considers $x^2\ln \left(x\right)$ to be $\mathrm{O}\left(x^2\right)$.

If a=infinity or a=-infinity, respectively, then the series expansion is only guaranteed to be valid for positive real x or negative real x, respectively. Use the substitution $x=\frac{1}{x}$ and then call series with $a=0$ to get an expansion that is valid around the North pole of the Riemann sphere.

Usually, the result of the series function is represented in the form of a series data structure. For an explanation of the data structure, see the type/series help page. However, the result of the series function will be represented in ordinary sum-of-products form, rather than in a series data structure, if it is a generalized series requiring fractional exponents, or if it is a series at $\pm \mathrm{\infty }$.

series implements a caching mechanism (see also option remember), so that when the same series is requested a second time but with a lower order, instead of recomputing it the terms from the cache are used.

$\mathrm{series}\left(\frac{x}{1-x-x^2}\&comma;x=0\right)$

$x+x^2+2x^3+3x^4+5x^5+O\left(x^6\right)$

$\mathrm{series}\left(\frac{x}{1-x-x^2}\&comma;x\right)$

$x+x^2+2x^3+3x^4+5x^5+O\left(x^6\right)$

$\mathrm{convert}\left(\&comma;\mathrm{polynom}\right)$

$5x^5+3x^4+2x^3+x^2+x$

$\mathrm{series}\left(x+\frac{1}{x}\&comma;x=1\&comma;3\right)$

$2+{\left(x-1\right)}^2+O\left({\left(x-1\right)}^3\right)$

$\mathrm{series}\left(\exp \left(x\right)\&comma;x=0\&comma;8\right)$

$1+x+\frac{1}{2}{x}^2+\frac{1}{6}{x}^3+\frac{1}{24}{x}^4+\frac{1}{120}{x}^5+\frac{1}{720}{x}^6+\frac{1}{5040}{x}^7+O\left(x^8\right)$

$\mathrm{series}\left(\frac{\exp \left(x\right)}{x}\&comma;x=0\&comma;8\right)$

$x^{\mathrm{-1}}+1+\frac{1}{2}x+\frac{1}{6}{x}^2+\frac{1}{24}{x}^3+\frac{1}{120}{x}^4+\frac{1}{720}{x}^5+\frac{1}{5040}{x}^6+O\left(x^7\right)$

$\mathrm{series}\left(\mathrm{\Gamma }\left(x\right)\&comma;x=0\&comma;2\right)$

$x^{\mathrm{-1}}-\mathrm{\gamma }+\left(\frac{{\mathrm{\pi }}^2}{12}+\frac{{\mathrm{\gamma }}^2}{2}\right)x+O\left(x^2\right)$

$\mathrm{series}\left(\frac{1}{\exp \left(x\right)-1-x}\&comma;x\&comma;6\right)$

$2x^{\mathrm{-2}}-\frac{2}{3}{x}^{\mathrm{-1}}+\frac{1}{18}+\frac{1}{270}x+O\left(x^2\right)$

$\mathrm{series}\left(\frac{1}{\exp \left(x\right)-1-x}\&comma;x\&comma;10\right)$

$2x^{\mathrm{-2}}-\frac{2}{3}{x}^{\mathrm{-1}}+\frac{1}{18}+\frac{1}{270}x-\frac{1}{3240}{x}^2-\frac{1}{13608}{x}^3-\frac{1}{2041200}{x}^4+\frac{1}{874800}{x}^5+O\left(x^6\right)$

$\mathrm{series}\left(\frac{1}{\exp \left(x\right)-1-x}\&comma;x\&comma;6\right)$

$2x^{\mathrm{-2}}-\frac{2}{3}{x}^{\mathrm{-1}}+\frac{1}{18}+\frac{1}{270}x-\frac{1}{3240}{x}^2-\frac{1}{13608}{x}^3-\frac{1}{2041200}{x}^4+\frac{1}{874800}{x}^5+O\left(x^6\right)$

$\mathrm{forget}\left(\mathrm{series}\right)\&colon;$

$\mathrm{series}\left(\frac{1}{\exp \left(x\right)-1-x}\&comma;x\&comma;6\right)$

$2x^{\mathrm{-2}}-\frac{2}{3}{x}^{\mathrm{-1}}+\frac{1}{18}+\frac{1}{270}x+O\left(x^2\right)$

$\mathrm{int}\left(\exp \left(x^3\right)\&comma;x\right)$

$-\frac{{\left(\mathrm{-1}\right)}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left(\frac{2x{\left(\mathrm{-1}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\mathrm{\pi }\sqrt{3}}{3\mathrm{\Gamma }\left(\frac{2}{3}\right){\left(-x^3\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}-\frac{x{\left(\mathrm{-1}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\mathrm{\Gamma }\left(\frac{1}{3}\&comma;-x^3\right)}{{\left(-x^3\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}\right)}{3}$

$\mathrm{series}\left(\&comma;x=0\right)$

$x+\frac{1}{4}{x}^4+O\left(x^7\right)$

$p≔\mathrm{series}\left(x^x\&comma;x=0\&comma;3\right)$

$p≔1+\ln \left(x\right)x+\frac{1}{2}{\ln \left(x\right)}^2{x}^2+O\left(x^3\right)$

$\mathrm{type}\left(p\&comma;'\mathrm{series}'\right)$

$\mathrm{true}$

$s≔\mathrm{series}\left(\mathrm{sqrt}\left(\sin \left(x\right)\right)\&comma;x=0\&comma;4\right)$

$s≔\sqrt{x}-\frac{x^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{12}+\mathrm{O}\left(x^{\raisebox{1ex}{$9$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)$

$\mathrm{type}\left(s\&comma;'\mathrm{series}'\right)$

$\mathrm{false}$

$\mathrm{whattype}\left(s\right)$

$\mathrm{`+`}$

$t≔\mathrm{series}\left(\frac{x^3}{x^4+4x-5}\&comma;x=\mathrm{\infty }\right)$

$t≔\frac{1}{x}-\frac{4}{x^4}+\frac{5}{x^5}+\mathrm{O}\left(\frac{1}{x^7}\right)$

$\mathrm{whattype}\left(t\right)$

$\mathrm{`+`}$

The series command was updated in Maple 2016; see Advanced Math.