Taylor Series Calculator
Compute the Taylor series expansion of f(x) about a given point up to degree n.
About Taylor Series Calculator
Computes the Taylor series T(x) = Σ f⁽ᵏ⁾(a)/k! · (x−a)ᵏ for k=0 to n. The coefficients f⁽ᵏ⁾(a) are estimated numerically using finite differences. The result is the polynomial approximation evaluated at the requested x, along with a comparison to the exact function value.
Formula
Taylor series
T(x) = Σ [f^(k)(a) / k!] · (x - a)^k, for k = 0 to n
- f^(k)(a)
- k-th derivative at center a
- k!
- k factorial
- a
- Center of expansion
- n
- Degree of polynomial
Worked example
With the default values below, Taylor Series Calculator returns T(x) at x of 0.3675125597. Change any input above and every figure here is recalculated in your browser.
Inputs used
- f(x)
- exp(x)
- Center (a)
- 0
- Degree (n)accepts 1–20
- 6
- Evaluate at x
- 1
Results
- T(x) at x
- 0.3675125597
- f(x) exact
- 2.7182818285
- Absolute error
- 2.3508 × 10⁰
- Coefficients
- c0=1, c1=-1, c2=0.5, c3=-0.166667, c4=0.041666, c5=-0.00824, c6=0.000753
Step by step
- 1Values used
- 2Taylor series
- 3T(x) at x= 0.3675125597
- 4f(x) exact= 2.7182818285
- 5Absolute error= 2.3508
- 6Coefficients= c0=1, c1=-1, c2=0.5, c3=-0.166667, c4=0.041666, c5=-0.00824, c6=0.000753
Assumptions and limits
- The result is derived only from the 4 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 10 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
How accurate is the Taylor approximation?
Accuracy improves with higher degree and proximity to the center a. The error is bounded by the (n+1)-th derivative term.
What is the difference between Taylor and Maclaurin series?
A Maclaurin series is simply a Taylor series centered at a = 0.