5.34.3 Series expansion : series
series takes from one to four arguments :
-
an expression depending of a variable (by default x),
- an equality variable=value (e.g. x=a) where to compute
the series expansion, by default x=0,
- an integer n, the order of the series expansion,
by default 5
- a direction -1, 1 (for unidirectional series expansion)
or 0 (for bidirectional series expansion) (by default 0).
Note that the syntax ...,x,a,n,...
(instead of ...,x=a,n,...) is also accepted.
series returns a polynomial in x-a, plus a remainder
of the form:
(x-a)^
n*order_size(x-a)
where order_size is a function such that,
∀ r>0, | | xr order_size(x) = 0 |
The order returned by series may be smaller than n if
cancellations between numerator and denominator occur.
Examples :
-
series expansion in the vicinity of x=0
Find an series expansion of
x3+sin(x)3/x−sin(x)
in the vicinity of x=0.
Input :
series(x^
3+sin(x)^
3/(x-sin(x)))
Output is only a 2nd-order series expansion :
6+-27/10*x^2
+x^
3*order_size(x)
We have lost 3 orders because the valuation of the numerator and
denominator is 3. To get a 4-th order expansion, we must therefore
take n=7.
Input :
series(x^
3+sin(x)^
3/(x-sin(x)),x=0,7)
or :
series(x^
3+sin(x)^
3/(x-sin(x)),x,0,7)
Output is a 4th-order series expansion :
6+-27/10*x^
2+x^
3+711/1400*x^
4+
x^
5*order_size(x)
- series expansion in the vicinity of x=a
Find a series 4th-order expansion of cos(2x)2 in the vicinity of
x=π/6.
Input:
series(cos(2*x)^
2,x=pi/6, 4)
Output :
1/4+(-(4*sqrt(3)))/4*(x-pi/6)+(4*3-4)/4*(x-pi/6)^
2+ 32*sqrt(3)/3/4*(x-pi/6)^
3+(-16*3+16)/3/4*(x-pi/6)^
4+ (x-pi/6)^
5*order_size(x-pi/6)
- series expansion in the vicinity of x=+∞ or x=-∞
-
Find a 5th-order series expansion of arctan(x) in the vicinity of
x=+∞.
Input :
series(atan(x),x=+infinity,5)
Output :
pi/2-1/x+1/3*(1/x)^
3+1/-5*(1/x)^
5+
(1/x)^
6*order_size(1/x)
Note that the expansion variable and the argument of the
order_size function is
h=1/x →x→ + ∞ 0 .
- Find a series 2nd-order expansion of (2x−1)e1/x−1 in the vicinity of
x=+∞.
Input :
series((2*x-1)*exp(1/(x-1)),x=+infinity,3)
Output is only a 1st-order series expansion :
2*x+1+2/x+(1/x)^
2*order_size(1/x)
To get a 2nd-order series expansion in 1/x, input:
series((2*x-1)*exp(1/(x-1)),x=+infinity,4)
Output :
2*x+1+2/x+17/6*(1/x)^
2+(1/x)^
3*order_size(1/x)
- Find a 2nd-order series expansion of (2x−1)e1/x−1 in the vicinity
of x=-∞.
Input :
series((2*x-1)*exp(1/(x-1)),x=-infinity,4)
Output :
-2*(-x)+1-2*(-1/x)+17/6*(-1/x)^
2+
(-1/x)^
3*order_size(-1/x)
- unidirectional series expansion.
The fourth parameter indicates the direction :
-
1 to do an series expansion in the vicinity of x=a with
x>a,
- -1 to do an series expansion in the vicinity of x=a with
x<a,
- 0 to do an series expansion in the vicinity of x=a with
x ≠ a.
For example,
find a 2nd-order series expansion of (1+x)1/x/x3 in
the vicinity of x=0+.
Input :
series((1+x)^
(1/x)/x^
3,x=0,2,1)
Output :
exp(1)/x^
3+(-(exp(1)))/2/x^
2+1/x*order_size(x)