> series(n +-> 1^n / factorial(n), x = 0)
Note that with this input your output type will be
UnivariatePuiseuxSeries(Expression(Integer),x,0).
Not bad, but you can have more control over the result type if you like.
Then, however, you have have to make sure that the interpreter does not
need to guess what you actually mean.
I demonstrate this now for
> series(n +-> %i^n / factorial(n), x = 0)
(1) -> Z==>Integer;Q==>Fraction Z;C==>Complex Q;S==>Stream C
Type:
Void
(2) -> n:=1
(2) 1
Type:
PositiveInteger
(3) -> nextCoefficient(c: C): C == (r:C := imaginary()*c/n; n:=n+1; r)
Function declaration nextCoefficient : Complex(Fraction(Integer))
-> Complex(Fraction(Integer)) has been added to workspace.
Type:
Void
(4) -> s: S := generate(nextCoefficient, 1$C)
Compiling function nextCoefficient with type Complex(Fraction(
Integer)) -> Complex(Fraction(Integer))
1 1 1 1 1 1 1 1
(4) [1,%i,- -,- - %i,--,--- %i,- ---,- ---- %i,-----,------ %i,...]
2 6 24 120 720 5040 40320 362880
Type:
Stream(Complex(Fraction(Integer)))
(5) -> series(s)$UnivariateTaylorSeries(C, x, 0)
(5)
1 2 1 3 1 4 1 5 1 6 1 7
1 + %i x - - x - - %i x + -- x + --- %i x - --- x - ---- %i x
2 6 24 120 720 5040
+
1 8 1 9 1 10 11
----- x + ------ %i x - ------- x + O(x )
40320 362880 3628800
Type:
UnivariateTaylorSeries(Complex(Fraction(Integer)),x,0)
UnivariateTaylorSeries does not itself allow a construction from a
coefficient function co: Z -> C.
But look at
https://github.com/hemmecke/fricas-svn/blob/master/src/algebra/genups.spad.pamphlet#L126
and in particular the auxiliary stream that is constructed there.
It should be possible to call "taylor" from there like
(20) -> taylor((n:Integer):Expression(Complex Integer) +-> %i^n /
factorial(n), x = 0)
(20)
1 2 %i 3 1 4 %i 5 1 6 %i 7 1 8
1 + %i x - - x - -- x + -- x + --- x - --- x - ---- x + ----- x
2 6 24 120 720 5040 40320
+
%i 9 1 10 11
------ x - ------- x + O(x )
362880 3628800
Type:
UnivariateTaylorSeries(Expression(Complex(Integer)),x,0)
or even
(21) -> series((n:Integer):Expression(Complex Integer) +-> %i^n /
factorial(n), x = 0)
(21)
1 2 %i 3 1 4 %i 5 1 6 %i 7 1 8
1 + %i x - - x - -- x + -- x + --- x - --- x - ---- x + ----- x
2 6 24 120 720 5040 40320
+
%i 9 1 10 11
------ x - ------- x + O(x )
362880 3628800
Type:
UnivariatePuiseuxSeries(Expression(Complex(Integer)),x,0)
However, look in particular at the result type, because it might not
exactly be that what you want for doing further computation.
Hope that helps.
Ralf
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