### 5.36.3 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,

lim
x→ 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=-∞
  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=1/x →x→ + ∞ 0 .
  2. 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)
  3. 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)