The function type/series returns true if the value of expr is Maple's series data structure, explained below.

The series data structure represents an expression as a truncated series in one specified indeterminate, expanded about a particular point. It is created by a call to the series function.

op(0, expr), with expr of type series, returns x-a where x denotes the ``series variable'' and a denotes the particular point of expansion. op(2*i-1, expr) returns the ith coefficient (a general expression) and op(2*i, expr) returns the corresponding integer exponent.

The exponents are ``word-size'' integers, in increasing order.

The representation is sparse; zero coefficients are not represented.

Usually, the final pair of operands in this data type are the special order symbol O(1) and the integer n which indicates the order of truncation. However, if the series is exact then there will be no order term, for example, the series expansion of a low-degree polynomial.

Formally, the coefficients of the series are such that

for some constants k1 and k2, for any $0<\mathrm{eps}$, and as x approaches a. In other words, the coefficients may depend on x, but their growth must be less than polynomial in x. O(1) represents such a coefficient, rather than an arbitrary constant.

A zero series is immediately simplified to the integer zero.

$a≔\mathrm{series}\left(\sin \left(x\right)\&comma;x=0\&comma;5\right)$

$a≔x-\frac{1}{6}{x}^3+O\left(x^5\right)$

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

$\mathrm{true}$

$\mathrm{type}\left(a\&comma;'\mathrm{taylor}'\right)$

$\mathrm{true}$

$\mathrm{op}\left(0\&comma;a\right)$

$x$

$\mathrm{op}\left(a\right)$

$1,1,-\frac{1}{6},3,\mathrm{O}\left(1\right),5$

$b≔\mathrm{series}\left(\frac{1}{\sin \left(x\right)}\&comma;x=0\&comma;5\right)$

$b≔x^{\mathrm{-1}}+\frac{1}{6}x+O\left(x^3\right)$

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

$\mathrm{true}$

$\mathrm{type}\left(b\&comma;'\mathrm{taylor}'\right)$

$\mathrm{false}$

$\mathrm{op}\left(0\&comma;b\right)$

$x$

$\mathrm{op}\left(b\right)$

$1,\mathrm{-1},\frac{1}{6},1,\mathrm{O}\left(1\right),3$

$\mathrm{type}\left(x^3\&comma;'\mathrm{series}'\right)$

$\mathrm{false}$

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

$\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)$

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

$\mathrm{false}$

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

$\mathrm{`+`}$

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

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

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

$\mathrm{true}$

$\mathrm{type}\left(c\&comma;'\mathrm{taylor}'\right)$

$\mathrm{false}$

$\mathrm{op}\left(0\&comma;c\right)$

$x$

$\mathrm{op}\left(c\right)$

$1,0,\ln \left(x\right),1,\frac{{\ln \left(x\right)}^2}{2},2,\mathrm{O}\left(1\right),3$

$d≔\mathrm{series}\left(\sin \left(x+y\right)\&comma;x=y\&comma;2\right)$

$d≔\sin \left(2y\right)+\cos \left(2y\right)\left(x-y\right)+O\left({\left(x-y\right)}^2\right)$

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

$\mathrm{true}$

$\mathrm{op}\left(0\&comma;d\right)$

$x-y$

$\mathrm{op}\left(d\right)$

$\sin \left(2y\right),0,\cos \left(2y\right),1,\mathrm{O}\left(1\right),2$