Taylor Series Explorer

Visualize Taylor and Maclaurin polynomial approximations

Function Selection

Choose Function:

Center Point (a):

Degree: 5

Original Function f(x)

Taylor Polynomial P_n(x)

Center Point x = a

General Form: Σ (-1)ⁿ x^(2n+1) / (2n+1)!

Maximum Absolute Error (on visible range):

0.0000

Current Polynomial P₅(x):

Taylor series coefficients: a_n = f⁽ⁿ⁾(a) / n!

The Taylor series of f(x) centered at a = 0 is:

f(x) = Σ [f⁽ⁿ⁾(a) / n!] (x - a)ⁿ

The n-th degree Taylor polynomial is the truncated series, which approximates f(x) near x = a.

As n increases, the approximation generally improves (within the radius of convergence).