Skip to content

Series

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

Series[f, {x, x0, n}]
    generates a power-series expansion of f about x = x0 up to order (x - x0)^n.
Series[f, x -> x0]
    generates the leading term of a power-series expansion of f about x = x0.
Series[f, {x, x0, nx}, {y, y0, ny}, ...]
    iteratively expands f, first in x, then in y, etc.
Series handles Taylor, Laurent (negative powers), and Puiseux (fractional powers)
    expansions, as well as logarithmic and symbolic-exponent cases such as x^x
    and (1+x)^n.
Series[f, {x, Infinity, n}] expands around x = Infinity by substituting x -> 1/u.
The result of Series is a SeriesData object; use Normal to convert it back to
    an ordinary expression by dropping the O-term.
Series is Protected and HoldAll so the expansion variable is not evaluated.

Examples

All examples below are verified against the current Mathilda build.

In[1]:= Series[Exp[x], {x, 0, 10}]
Out[1]= 1 + x + 1/2 x^2 + 1/6 x^3 + 1/24 x^4 + 1/120 x^5 + 1/720 x^6 + 1/5040 x^7 + 1/40320 x^8 + 1/362880 x^9 + 1/3628800 x^10 + O[x]^11

In[2]:= Series[f[x], {x, a, 3}]
Out[2]= f[a] + Derivative[1][f][a] (x - a) + 1/2 Derivative[2][f][a] (x - a)^2 + 1/6 Derivative[3][f][a] (x - a)^3 + O[x - a]^4

In[3]:= Series[Cos[x]/x, {x, 0, 10}]
Out[3]= 1/x - 1/2 x + 1/24 x^3 - 1/720 x^5 + 1/40320 x^7 - 1/3628800 x^9 + O[x]^11

In[4]:= Series[Sqrt[Sin[x]], {x, 0, 10}]
Out[4]= Sqrt[x] - 1/12 x^(5/2) + 1/1440 x^(9/2) - 1/24192 x^(13/2) - 67/29030400 x^(17/2) + O[x]^(21/2)

In[5]:= Series[x^x, {x, 0, 4}]
Out[5]= 1 + Log[x] x + 1/2 Log[x]^2 x^2 + 1/6 Log[x]^3 x^3 + 1/24 Log[x]^4 x^4 + O[x]^5

In[6]:= Series[(1 + x)^n, {x, 0, 4}]
Out[6]= 1 + n x + 1/2 n (-1 + n) x^2 + 1/6 n (-2 + n) (-1 + n) x^3 + 1/24 n (-3 + n) (-2 + n) (-1 + n) x^4 + O[x]^5

In[7]:= Series[Sin[1/x], {x, Infinity, 10}]
Out[7]= 1/x - 1/6 (1/x)^3 + 1/120 (1/x)^5 - 1/5040 (1/x)^7 + 1/362880 (1/x)^9 + O[1/x]^11

In[8]:= Series[Sin[x + y], {x, 0, 3}, {y, 0, 3}]
Out[8]= y - 1/6 y^3 + O[y]^4 + (1 - 1/2 y^2 + O[y]^4) x + (-1/2 y + 1/12 y^3 + O[y]^4) x^2 + (-1/6 + 1/12 y^2 + O[y]^4) x^3 + O[x]^4

Implementation notes

Algorithm. Series computes a truncated power series via recursive series algebra. builtin_series (ATTR_HOLDALL) parses each spec — full form {x, x0, n} or leading-term form x -> x0 (via parse_series_spec) — threads a List first argument, and handles the multivariate form by expanding left-to- right (each coefficient of the outer series is recursively expanded in the next variable). Inexact inputs are rationalised then numericalised back (internal_rationalize_then_numericalize).

do_series_single does the work. It evaluates f, returns it verbatim if free of x, optionally Apart-decomposes a rational function in x (each partial-fraction term hits the cheap monomial binomial path instead of a Newton inversion), pulls out a symbolic x^alpha prefactor when expanding at 0 or Infinity, and handles expansion at Infinity by substituting x -> 1/u. It expands to a padded internal order (order + pad, pad = 12 for numeric x0, 2 for symbolic x0 to keep symbolic convolution from exploding) and then truncates back to the user O-term; the leading-term form widens the O-term to the first non-zero coefficient.

The recursion is series_expand(e, ctx): a subexpression free of x becomes a constant series; x itself becomes the identity series; Plus/Times/Power combine child series with so_add/so_mul/so_inv/so_pow_int; the elementary heads Exp, Log, Sin, Cos, Tan, Sinh, Cosh, Tanh and the inverse trig/hyperbolic family compose their known Taylor kernels (kernel_coefs → so_compose_scalar_kernel) with the inner argument's series, with reciprocal heads (Sec/Csc/Cot/...) rewritten via rewrite_reciprocal_head and a large set of branch-point / at-infinity identities for the inverse functions. Any unrecognised head falls back to naive Taylor series_taylor_via_D: coefficients a_k = (D^k f at x0)/k!, capped at MAX_NAIVE_ORDER = 20 and bailing out (has_infinity) at singularities.

Data structures. The internal SeriesObj struct holds the expansion variable x, expansion point x0, an owned array of coefficient Expr*, the leading exponent numerator nmin, the (exclusive) O-term exponent numerator order, and a common exponent denominator den (>= 1) — so coefficient i multiplies (x - x0)^((nmin+i)/den), supporting Laurent and Puiseux (fractional-exponent) series. so_rescale/so_align_den reconcile denominators before arithmetic. SeriesObj is converted to the user-facing SeriesData[x, x0, {coefs}, nmin, nmax, den] head by so_to_expr.

Complexity / limits. Kernel-path heads are exact and fast; the naive D-path is capped at order 20 and fails at true branch points where derivatives blow up (Puiseux at such points is out of scope except for the explicit inverse-function branch-point handlers). Symbolic expansion points cap the internal pad tightly to avoid O(N^2) symbolic coefficient blow-up.

  • HoldAll and Protected (so the expansion variable is not evaluated before Series has a chance to shield it).
  • Threaded over lists: Series[{f1, f2, ...}, spec] becomes {Series[f1, spec], Series[f2, spec], ...}.
  • Handles Taylor expansions for smooth functions, Laurent expansions where the function has a pole at x0, Puiseux expansions for fractional-power cases such as Sqrt[Sin[x]], and logarithmic expansions for cases like x^x where Log[x] survives as a symbolic coefficient.
  • Symbolic parameters in exponents are supported: Series[(1 + x)^n, {x, 0, 4}] returns the binomial expansion with n kept unexpanded.
  • Approximate numeric coefficients flow through series arithmetic unchanged.
  • For unknown heads (e.g. f[x] where f has no rules), the engine falls back to naive Taylor via D at the expansion point; the coefficients appear as Derivative[k][f][x0].
  • The Assumptions -> assm option selects the branch of the logarithmic expansions at x = 0 (ExpIntegralEi, LogIntegral). When assm forces the expansion variable negative (e.g. x < 0, x < -2, 0 > x, or an And[...] containing such a relation), Log[x] is emitted as Log[-x]; otherwise the principal x > 0 form is used. The option is matched by its LHS symbol Assumptions (so it is not confused with a leading-term spec x -> x0) and is forwarded into each inner variable for multivariate expansions.

Attributes: HoldAll, Protected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References

Notes & additional examples

Worked examples

In[1]:= Series[Sin[x], {x, 0, 5}]
Out[1]= x - 1/6 x^3 + 1/120 x^5 + O[x]^6

In[1]:= Series[1/(1 - x), {x, 0, 4}]
Out[1]= 1 + x + x^2 + x^3 + x^4 + O[x]^5

In[1]:= Series[Log[1 + x], {x, 0, 4}]
Out[1]= x - 1/2 x^2 + 1/3 x^3 - 1/4 x^4 + O[x]^5

In[1]:= Normal[Series[Exp[x], {x, 0, 3}]]
Out[1]= 1 + x + 1/2 x^2 + 1/6 x^3

The expansion of Tan recovers the tangent numbers as coefficients:

In[1]:= Series[Tan[x], {x, 0, 7}]
Out[1]= x + 1/3 x^3 + 2/15 x^5 + 17/315 x^7 + O[x]^8

A Laurent expansion appears automatically when f has a pole at the base point; 1/(E^x - 1) starts at x^(-1) and its coefficients are the Bernoulli numbers over factorials:

In[1]:= Series[1/(Exp[x] - 1), {x, 0, 4}]
Out[1]= 1/x - 1/2 + 1/12 x - 1/720 x^3 + O[x]^5

Series also handles symbolic-exponent expansions such as x^x, where the coefficients involve Log[x]:

In[1]:= Series[x^x, {x, 0, 3}]
Out[1]= 1 + Log[x] x + 1/2 Log[x]^2 x^2 + 1/6 Log[x]^3 x^3 + O[x]^4

Expansion at infinity recovers the classic limit (1 + 1/x)^x -> E, with the correction terms made explicit:

In[1]:= Series[(1 + 1/x)^x, {x, Infinity, 2}]
Out[1]= E + (-1/2 E)/x + 11/24 E (1/x)^2 + O[1/x]^3

Notes

Series[f, {x, x0, n}] builds a power-series expansion to order (x - x0)^n, returning a SeriesData object that prints with a trailing O[x]^(n+1) term. It handles Taylor, Laurent (negative powers), and Puiseux (fractional powers) cases, as well as expansion around Infinity via the x -> 1/u substitution. Apply Normal to drop the order term and recover an ordinary polynomial, as in the Exp example above. Series is HoldAll, so the expansion variable is held unevaluated while the expansion point and order are read off.