### 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

taylor(sin(x),x=1,2)

or:

series(sin(x),x=1,2)

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

taylor(sin(x),x,2,1)

or:

series(sin(x),x,2,1)

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 consider

x3+sin(x)3
x−sin(x)

.

taylor(x^3+sin(x)^3/(x-sin(x)),x=0,5)

6−

27
10

x2+x3+

711
1400

x4+x6 order_size

⎛
⎝

x

⎞
⎠

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.

taylor(x^3+sin(x)^3/(x-sin(x)),x=0,7)

6−

27
10

x2+x3+

711
1400

x4−

737
14000

x6+x8 order_size

⎛
⎝

x

⎞
⎠

a fourth degree expansion.

#### Examples

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

taylor(cos(2*x)^2,x=pi/6, 4)

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=+∞.

series(atan(x),x=+infinity,5)

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 → 0 as x→+∞.

Find a 2nd-order expansion of (2x−1)e1/x−1 in the vicinity of x=+∞.

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:

series((2*x-1)*exp(1/(x-1)),x=+infinity,4)

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)e1/x−1 in the vicinity of x=-∞.

series((2*x-1)*exp(1/(x-1)),x=-infinity,4)

−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+.

series((1+x)^(1/x)/x^3,x=0,2,1)

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

e x−3−

e
2

x−2+x−1 order_size

⎛
⎝

x

⎞
⎠