Taylor Series Calculator
Compute the Taylor series expansion of f(x) about a given point up to degree n.
Inputs
Enter function using x.
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
Values used
f(x) = exp(x); Center (a) = 0; Degree (n) = 6; Evaluate at x = 1
Taylor series
T(x) = Σ [f^(k)(a) / k!] · (x - a)^k, for k = 0 to n
T(x) at x
= 0.3675125597
f(x) exact
= 2.7182818285
Absolute error
= 2.3508
Coefficients
= c0=1, c1=-1, c2=0.5, c3=-0.166667, c4=0.041666, c5=-0.00824, c6=0.000753
How it works
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
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.