Series expansion :

- 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`).

`^`

n*order_size(x-a)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)))`6+-27/10*x``^2`

+x`^`

3*order_size(x)*n*= 7, input:`series(x``^`

3+sin(x)`^`

3/(x-sin(x)),x=0,7)`series(x``^`

3+sin(x)`^`

3/(x-sin(x)),x,0,7)`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(2*x*)^{2}in the vicinity of*x*= .

Input:`series(cos(2*x)``^`

2,x=pi/6, 4)`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)``pi/2-1/x+1/3*(1/x)``^`

3+1/-5*(1/x)`^`

5+ (1/x)`^`

6*order_size(1/x)`order_size`function is*h*= 0. - Find a series 2nd-order expansion of
(2
*x*- 1)*e*^{}1*x*-1 in the vicinity of`x=+`.

Input :`series((2*x-1)*exp(1/(x-1)),x=+infinity,3)``2*x+1+2/x+(1/x)``^`

2*order_size(1/x)*x*, input:`series((2*x-1)*exp(1/(x-1)),x=+infinity,4)``2*x+1+2/x+17/6*(1/x)``^`

2+(1/x)`^`

3*order_size(1/x) - Find a 2nd-order series expansion of
(2
*x*- 1)*e*^{}1*x*-1) in the vicinity of`x=-`.

Input :`series((2*x-1)*exp(1/(x-1)),x=-infinity,4)``-2*(-x)+1-2*(-1/x)+17/6*(-1/x)``^`

2+

(-1/x)`^`

3*order_size(-1/x)

- Find a 5th-order series expansion of
arctan(
- 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*.

*x*= 0^{+}. Input :`series((1+x)``^`

(1/x)/x`^`

3,x=0,2,1)`exp(1)/x``^`

3+(-(exp(1)))/2/x`^`

2+1/x*order_size(x)