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## Series expansion : series

series takes from one to four arguments :
• an expression dependending 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,

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 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 = .
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=-
1. 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 = 0.
2. Find a series 2nd-order expansion of (2x - 1)e1x-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)
3. Find a 2nd-order series expansion of (2x - 1)e1x-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    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)

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giac documentation written by Renée De Graeve and Bernard Parisse