Taylor Series Calculator
Find Taylor and Maclaurin polynomials step by step. Enter a supported function, choose the center and degree, and optionally evaluate the approximation at a target value of x. The calculator shows the polynomial, the derivative pattern, approximation results, and clear student-friendly explanations.
Background
A Taylor polynomial approximates a function near a chosen center a using derivatives. When a = 0, the result is called a Maclaurin polynomial. These approximations are extremely useful in Calculus because polynomials are usually easier to evaluate, differentiate, and integrate than the original function.
How to use this calculator
- Choose Taylor or Maclaurin mode.
- Enter a supported function such as e^x, sin(x), ln(1+x), ln(x), 1/(1-x), or a polynomial like x^3-2x+1.
- Choose the center a and the degree n.
- Optionally enter a target value of x to evaluate the polynomial approximation.
- Click Calculate to see the polynomial, derivative pattern, approximation details, and the step-by-step construction.
How this calculator works
- It uses the Taylor polynomial formula T_n(x) = Σ [f^(k)(a) / k!] (x-a)^k.
- The calculator finds the needed coefficients up to your chosen degree.
- For common functions, it uses known derivative patterns or standard series forms.
- If you enter an evaluation point, it computes the polynomial value there and compares it to the actual function value whenever the input is supported for direct evaluation.
- The approximation is generally best when x stays close to the center a.
Formula & Equations Used
Taylor polynomial: T_n(x) = Σ [f^(k)(a)/k!] (x-a)^k
Maclaurin polynomial: T_n(x) = Σ [f^(k)(0)/k!] x^k
Approximation idea: f(x) ≈ T_n(x) near the center.
Absolute error: |f(x) - T_n(x)|
Example Problem & Step-by-Step Solution
Example 1 — Degree-4 Maclaurin polynomial for e^x
Since every derivative of e^x is still e^x, all derivative values at x = 0 equal 1.
- Use T_4(x) = f(0) + f'(0)x + f''(0)x²/2! + f'''(0)x³/3! + f⁽⁴⁾(0)x⁴/4!.
- Substitute the derivative values 1,1,1,1,1.
- Get T_4(x) = 1 + x + x²/2 + x³/6 + x⁴/24.
Example 2 — Degree-3 Taylor polynomial for ln(x) centered at a = 1
- Compute derivatives: f'(x)=1/x, f''(x)=-1/x², f'''(x)=2/x³.
- Evaluate at x = 1: f(1)=0, f'(1)=1, f''(1)=-1, f'''(1)=2.
- Build the polynomial: T_3(x) = (x-1) - (x-1)²/2 + (x-1)³/3.
Example 3 — Approximate e^{0.2}
Using T_3(x)=1+x+x²/2+x³/6, T_3(0.2)=1+0.2+0.02+0.001333…≈1.221333, which is very close to the actual value.
Frequently Asked Questions
Q: What is the difference between a Taylor series and a Maclaurin series?
A Maclaurin series is just a Taylor series centered at a = 0.
Q: Why does the approximation work best near the center?
Because the polynomial is built from the function and its derivatives at that center, so the match is usually strongest nearby.
Q: Does a higher degree always help?
Usually near the center, yes. But farther away from the center, even a higher-degree polynomial may lose accuracy.
Q: Why do factorials appear in the formula?
Each derivative term is divided by n! so the polynomial is scaled correctly term by term.
Q: Why is arctan(x) limited to Maclaurin here?
This version focuses on the most common classroom use and keeps the calculator fast, reliable, and student-friendly.
