Taylor Series Calculator

Approximate functions using Taylor series expansion. Compare the approximation with the exact value for sin, cos, e^x, and ln(x).

Function

Center Point (a)

Number of Terms

Evaluate at x

Tab to move between fields

Taylor Approximation

0.83333

Error0.008138

Last Term Coefficient-0

How to Use This Calculator

Select the function to approximate: sin(x), cos(x), e^x, or ln(1+x).
Enter the center point a (Taylor series expands around this point; Maclaurin series uses a = 0).
Set the number of terms to include; more terms extend the interval where the approximation is accurate.
The Taylor polynomial is built from successive derivatives: f(a) + f'(a)(x−a) + f''(a)(x−a)²/2! + …
The chart overlays the exact function and the Taylor polynomial so you can see how closely they agree near x = a.
Increasing the number of terms improves accuracy but the series still diverges far from the center for some functions (e.g., ln(1+x) for |x| > 1).

How the result changes with Evaluate at x

Identify Input Parameters
4 parameters
Function = 1, Center Point (a) = 0, Number of Terms = 5, Evaluate at x = 1 = 4 input(s) provided
Calculate Taylor Approximation
Taylor Approximation
0.83333 = 0.83333
Calculate Error
Error
0.008138 = 0.008138
Calculate Last Term Coefficient
Last Term Coefficient
0 = 0

Engine last updated August 5, 2026. Checked against 3 independently-derived tests — how we verify calculators.

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