series data structure
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<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 ≔ series⁡sin⁡x,x=0,5
b ≔ series⁡1sin⁡x,x=0,5
c ≔ series⁡xx,x=0,3
d ≔ series⁡sin⁡x+y,x=y,2
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