Taylor Series
Taylor and Maclaurin polynomial approximation of a function
About this calculator
The Taylor Series Calculator builds the Taylor or Maclaurin polynomial of f(x) about a center a up to order 12, computing every coefficient cₖ = f⁽ᵏ⁾(a)/k! by power-series automatic differentiation — exact recurrences on the expression tree, far more stable than finite differences. It lists each derivative and coefficient, prints the polynomial in powers of (x − a), and can evaluate Pₙ(x) at a point to compare against the true f(x) with the absolute truncation error. Singular centers and non-analytic functions are detected and reported.
How to use the Taylor Series calculator
- Enter the values for your problem into the input fields.
- Read the result — it updates instantly as you type.
- Check the formula and the visual explanation to follow how the answer was found.
- Copy the page URL to share the exact calculation.
Common examples
- sin(x) about 0, order 7 → x − 0.166667x³ + 0.00833333x⁵ − 0.000198413x⁷
- exp(x), order 4, evaluated at x = 1 → P₄(1) = 2.70833… vs e = 2.71828… (error ≈ 0.00995)
- ln(x) about a = 1, order 4 → (x − 1) − 0.5(x − 1)² + 0.333333(x − 1)³ − 0.25(x − 1)⁴
- 1/(1 − x) about 0, order 5 → 1 + x + x² + x³ + x⁴ + x⁵, the geometric series
- sin(x), order 5, at x = 0.5 → P₅(0.5) = 0.479427… vs 0.479426…, error ≈ 1.5×10⁻⁶
Frequently asked questions
How does the Taylor Series calculator work?
Enter your values and the calculator applies the exact mathematical method for this problem, showing the result together with the formula it used. Everything is computed with high-precision arithmetic, so the answer you see is not limited by ordinary floating-point rounding.
When would I use the Taylor Series calculator?
It is useful for homework and exam preparation, for checking work done by hand, and whenever a step in a larger problem needs this calculation done quickly and reliably.
How accurate are the results?
Calculations run at 30 significant digits of precision internally. Displayed values are rounded for readability, but the underlying result is far more precise than a typical hand or pocket-calculator computation.
Can I share a specific calculation with someone else?
Yes. Every value you enter updates the URL, so copying the address bar and sharing the link reopens the exact same calculation for whoever clicks it.