series - Maple Help

type/series

series data structure

 Calling Sequence type(expr, series)

Parameters

 expr - expression

Description

 • 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

$\mathrm{k1}{\left(x-a\right)}^{\mathrm{eps}}<|{\mathrm{coeff}}_{i}|<\frac{\mathrm{k2}}{{\left(x-a\right)}^{\mathrm{eps}}}$

 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.

Examples

 > $a≔\mathrm{series}\left(\mathrm{sin}\left(x\right),x=0,5\right)$
 ${a}{≔}{x}{-}\frac{{1}}{{6}}{}{{x}}^{{3}}{+}{O}{}\left({{x}}^{{5}}\right)$ (1)
 > $\mathrm{type}\left(a,'\mathrm{series}'\right)$
 ${\mathrm{true}}$ (2)
 > $\mathrm{type}\left(a,'\mathrm{taylor}'\right)$
 ${\mathrm{true}}$ (3)
 > $\mathrm{op}\left(0,a\right)$
 ${x}$ (4)
 > $\mathrm{op}\left(a\right)$
 ${1}{,}{1}{,}{-}\frac{{1}}{{6}}{,}{3}{,}{\mathrm{O}}{}\left({1}\right){,}{5}$ (5)
 > $b≔\mathrm{series}\left(\frac{1}{\mathrm{sin}\left(x\right)},x=0,5\right)$
 ${b}{≔}{{x}}^{{-1}}{+}\frac{{1}}{{6}}{}{x}{+}{O}{}\left({{x}}^{{3}}\right)$ (6)
 > $\mathrm{type}\left(b,'\mathrm{series}'\right)$
 ${\mathrm{true}}$ (7)
 > $\mathrm{type}\left(b,'\mathrm{taylor}'\right)$
 ${\mathrm{false}}$ (8)
 > $\mathrm{op}\left(0,b\right)$
 ${x}$ (9)
 > $\mathrm{op}\left(b\right)$
 ${1}{,}{-1}{,}\frac{{1}}{{6}}{,}{1}{,}{\mathrm{O}}{}\left({1}\right){,}{3}$ (10)
 > $\mathrm{type}\left({x}^{3},'\mathrm{series}'\right)$
 ${\mathrm{false}}$ (11)
 > $\mathrm{series}\left(\mathrm{sqrt}\left(\mathrm{sin}\left(x\right)\right),x=0,4\right)$
 $\sqrt{{x}}{-}\frac{{{x}}^{{5}}{{2}}}}{{12}}{+}{\mathrm{O}}{}\left({{x}}^{{9}}{{2}}}\right)$ (12)
 > $\mathrm{type}\left(,'\mathrm{series}'\right)$
 ${\mathrm{false}}$ (13)
 > $\mathrm{whattype}\left(\right)$
 ${\mathrm{+}}$ (14)
 > $c≔\mathrm{series}\left({x}^{x},x=0,3\right)$
 ${c}{≔}{1}{+}{\mathrm{ln}}{}\left({x}\right){}{x}{+}\frac{{1}}{{2}}{}{{\mathrm{ln}}{}\left({x}\right)}^{{2}}{}{{x}}^{{2}}{+}{O}{}\left({{x}}^{{3}}\right)$ (15)
 > $\mathrm{type}\left(c,'\mathrm{series}'\right)$
 ${\mathrm{true}}$ (16)
 > $\mathrm{type}\left(c,'\mathrm{taylor}'\right)$
 ${\mathrm{false}}$ (17)
 > $\mathrm{op}\left(0,c\right)$
 ${x}$ (18)
 > $\mathrm{op}\left(c\right)$
 ${1}{,}{0}{,}{\mathrm{ln}}{}\left({x}\right){,}{1}{,}\frac{{{\mathrm{ln}}{}\left({x}\right)}^{{2}}}{{2}}{,}{2}{,}{\mathrm{O}}{}\left({1}\right){,}{3}$ (19)
 > $d≔\mathrm{series}\left(\mathrm{sin}\left(x+y\right),x=y,2\right)$
 ${d}{≔}{\mathrm{sin}}{}\left({2}{}{y}\right){+}{\mathrm{cos}}{}\left({2}{}{y}\right){}\left({x}{-}{y}\right){+}{O}{}\left({\left({x}{-}{y}\right)}^{{2}}\right)$ (20)
 > $\mathrm{type}\left(d,'\mathrm{series}'\right)$
 ${\mathrm{true}}$ (21)
 > $\mathrm{op}\left(0,d\right)$
 ${x}{-}{y}$ (22)
 > $\mathrm{op}\left(d\right)$
 ${\mathrm{sin}}{}\left({2}{}{y}\right){,}{0}{,}{\mathrm{cos}}{}\left({2}{}{y}\right){,}{1}{,}{\mathrm{O}}{}\left({1}\right){,}{2}$ (23)