** suivant:** Series expansion : series
** monter:** Taylor and asymptotic expansions
** précédent:** Division by increasing power
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##

Taylor expansion : `taylor`

`taylor` 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 Taylor 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,`*n*,*a*,...
(instead of `...,x=`*a*,*n*,...) is also accepted.

`taylor` 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,
For regular series expansion, `order_size` is a bounded function,
but for non regular series expansion, it might tend slowly to
infinity, for example like a power of ln(*x*).

Input :
`taylor(sin(x),x=1,2)`

Or (be carefull with the order of the arguments !) :
`taylor(sin(x),x,2,1)`

Output :
`sin(1)+cos(1)*(x-1)+(-(1/2*sin(1)))*(x-1)``^`

2+ (x-1)`^`

3*order_size(x-1)

**Remark**

The order returned by `taylor` may be smaller than *n* if
cancellations between numerator and denominator occur, for example

taylor(

)

Input :
`taylor(x``^`

3+sin(x)`^`

3/(x-sin(x)))

The output is only a 2nd-order series expansion :
` 6+-27/10*x``^2`

+x`^`

3*order_size(x)

Indeed the numerator and denominator valuation is 3, hence we loose 3
orders. To get order 4, we should ask *n* = 7, input :
`taylor(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)

** suivant:** Series expansion : series
** monter:** Taylor and asymptotic expansions
** précédent:** Division by increasing power
** Table des matières**
** Index**
giac documentation written by Renée De Graeve and Bernard Parisse