13.5.2  Series expansion

The taylor or series command finds Taylor expansions.

• taylor takes one mandatory and four optional arguments:
  • expr, an expression depending on a variable.
  • Optionally, x, the variable (by default x).
  • Optionally n, an integer, the order of the series expansion (by default 5).
  • Optionally, a, the center of the Taylor expansion (by default 0). This can be combined with the optional x by replacing x by x=a.
  • dir, a direction, which can be -1 or 1, for unidirectional series expansion, or 0 (for bidirectional series expansion) (by default 0).
• taylor(expr,x ⟨,a,n,dir ⟩) returns the Taylor expansion of expr about a or order n; consisting of a polynomial in x−a plus a remainder of the form 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

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).

Example

or:

or (be careful with the order of the arguments):

or:

Remark.

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

which is only a 2^nd degree expansion. Indeed the numerator and denominator valuation is 3, hence you lose 3 orders. To get order 4, you should use n=7.

a fourth degree expansion.

Examples

Find a 4th-order expansion of cos(2x)² in the vicinity of x=π/6.

Find a 5th-order series expansion of arctan(x) in the vicinity of x=+∞.

Note that the expansion variable and the argument of the order_size function is h=1/x → 0 as x→+∞.

Find a 2^nd-order expansion of (2x−1)e^1/x−1 in the vicinity of x=+∞.

Output (only a 1^st-order series expansion):

Note that this is only a 1^st-order expansion. To get a 2^nd-order series expansion in 1/x:

Find a 2^nd-order series expansion of (2x−1)e^1/x−1 in the vicinity of x=-∞.

Find a 2^nd-order series expansion of (1+x)^1/x/x³ in the vicinity of x=0⁺.

(Note that this is a one-sided series expansion, since dir=1.)