6.36.2  Series expansion: taylor series

The taylor command finds Taylor expansions.
series is a synonym for taylor.

• 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,ndir⟩) 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.
Input:

or:

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

or:

Output:

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

Input:

Output:

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

Output:

a fourth degree expansion.

Examples.

• Find a 4th-order expansion of cos(2x)² in the vicinity of x=π/6.
  Input:
  taylor(cos(2*x)^2,x=pi/6, 4)
  Output:

  1 4

  −

  √

  3

  ⎛ ⎜ ⎜ ⎝

  x−

  π  6

  ⎞ ⎟ ⎟ ⎠

  +2

  ⎛ ⎜ ⎜ ⎝

  x−

  π  6

  ⎞ ⎟ ⎟ ⎠

  2

  +

  8 3

  √

  3

  ⎛ ⎜ ⎜ ⎝

  x−

  π  6

  ⎞ ⎟ ⎟ ⎠

  3

  −

  8 3

  ⎛ ⎜ ⎜ ⎝

  x−

  π  6

  ⎞ ⎟ ⎟ ⎠

  4

  +

  ⎛ ⎜ ⎜ ⎝

  x−

  π  6

  ⎞ ⎟ ⎟ ⎠

  5

  order_size

  ⎛ ⎜ ⎜ ⎝

  x−

  π  6

  ⎞ ⎟ ⎟ ⎠

• Find a 5th-order series expansion of arctan(x) in the vicinity of x=+∞.
  Input:
  series(atan(x),x=+infinity,5)
  Output:

  π  2

  −

  1 x

  +

  ⎛ ⎜ ⎜ ⎝ 1 x ⎞ ⎟ ⎟ ⎠ 3   3

  −

  ⎛ ⎜ ⎜ ⎝ 1 x ⎞ ⎟ ⎟ ⎠ 5   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 .
• Find a 2nd-order expansion of (2x−1)e^1/x−1 in the vicinity of x=+∞.
  Input:
  series((2*x-1)*exp(1/(x-1)),x=+infinity,3)
  Output (only a 1st-order series expansion):

  2

  ⎛ ⎜ ⎜ ⎝

  1 x

  ⎞ ⎟ ⎟ ⎠

  −1

  +1+

  2 x

  +

  17 6

  ⎛ ⎜ ⎜ ⎝

  1 x

  ⎞ ⎟ ⎟ ⎠

  2

  +

  ⎛ ⎜ ⎜ ⎝

  1 x

  ⎞ ⎟ ⎟ ⎠

  3

  order_size

  ⎛ ⎜ ⎜ ⎝

  1 x

  ⎞ ⎟ ⎟ ⎠

  Note that this is only a 1st order expansion. To get a 2nd-order series expansion in 1/x:
  Input:
  series((2*x-1)*exp(1/(x-1)),x=+infinity,4)
  Output:

  2

  ⎛ ⎜ ⎜ ⎝

  1 x

  ⎞ ⎟ ⎟ ⎠

  −1

  +1+

  2 x

  +

  17 6

  ⎛ ⎜ ⎜ ⎝

  1 x

  ⎞ ⎟ ⎟ ⎠

  2

  +

  47 12

  ⎛ ⎜ ⎜ ⎝

  1 x

  ⎞ ⎟ ⎟ ⎠

  3

  +

  ⎛ ⎜ ⎜ ⎝

  1 x

  ⎞ ⎟ ⎟ ⎠

  4

  order_size

  ⎛ ⎜ ⎜ ⎝

  1 x

  ⎞ ⎟ ⎟ ⎠

• Find a 2nd-order series expansion of (2x−1)e^1/x−1 in the vicinity of x=-∞.
  Input:
  series((2*x-1)*exp(1/(x-1)),x=-infinity,4)
  Output:

  −2

  ⎛ ⎜ ⎜ ⎝

  −

  1 x

  ⎞ ⎟ ⎟ ⎠

  −1

  +1+

  2 x

  +

  17 6

  ⎛ ⎜ ⎜ ⎝

  −

  1 x

  ⎞ ⎟ ⎟ ⎠

  2

  −

  47 12

  ⎛ ⎜ ⎜ ⎝

  −

  1 x

  ⎞ ⎟ ⎟ ⎠

  3

  +

  ⎛ ⎜ ⎜ ⎝

  −

  1 x

  ⎞ ⎟ ⎟ ⎠

  4

  order_size

  ⎛ ⎜ ⎜ ⎝

  −

  1 x

  ⎞ ⎟ ⎟ ⎠

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

  e x−3−

  1 2

  e x−2+x−1 order_size

  ⎛ ⎝

  x

  ⎞ ⎠