Power Series (Taylor Series) Calculator
Expand any function f(x) into its Taylor (power) series about a point a. Enter the function, expansion point, and number of terms to see each term and the assembled polynomial.
Updated 2026-06-14 · Free · No sign-up · Runs privately in your browser
| k | f⁽ᵏ⁾(a) | f⁽ᵏ⁾(a)/k! | Term |
|---|
Show the formula & steps
How the Power Series Calculator Works
This calculator expands a function f(x) into its Taylor series (a power series) about a chosen center a. Enter a function such as exp(x), sin(x), or 1/(1-x), pick the expansion point, choose how many terms you want, and it returns each term, the full polynomial, and — if you give an x value — the numerical approximation alongside the exact answer and the error.
When the center is a = 0, the result is the Maclaurin series, a special case of the Taylor series.
The Formula
T(x) = Σ (from k = 0 to n) f⁽ᵏ⁾(a) / k! · (x − a)ᵏ
Each term needs three ingredients: the k-th derivative of f, evaluated at the center a; divided by k factorial; multiplied by (x − a) raised to the k-th power.
The tool computes f and its derivatives symbolically, evaluates each at a, and assembles the sum.
Worked Example
Expand f(x) = eˣ about a = 0 with 5 terms.
Every derivative of eˣ is eˣ, and e⁰ = 1, so f⁽ᵏ⁾(0) = 1 for all k. The coefficients are therefore 1/k!:
| k | f⁽ᵏ⁾(0) | f⁽ᵏ⁾(0)/k! | Term |
|---|---|---|---|
| 0 | 1 | 1 | 1 |
| 1 | 1 | 1 | x |
| 2 | 1 | 0.5 | 0.5·x² |
| 3 | 1 | 0.166667 | 0.166667·x³ |
| 4 | 1 | 0.041667 | 0.041667·x⁴ |
So T(x) = 1 + x + x²/2 + x³/6 + x⁴/24.
Evaluating at x = 0.5 gives T(0.5) ≈ 1.6484, while the exact value e^0.5 ≈ 1.6487 — an error of only about 0.0003 with just five terms.
Why the Center Matters
A Taylor polynomial is most accurate near its center a and loses accuracy as x moves away. It also only converges within the function’s radius of convergence. For example, the series for 1/(1 − x) about a = 0 is 1 + x + x² + x³ + …, which converges only for |x| < 1; outside that interval the approximation diverges no matter how many terms you add.
Tips
- Write multiplication explicitly (
2*x, not2x) and use^for powers. - Built-in functions include
exp,sin,cos,tan,log(natural log),sqrt, and more. - Pick a center where f is defined. Expanding
log(x)about a = 0 fails because log(0) and its derivatives blow up — try a = 1 instead. - Increase the term count to shrink the error shown next to the exact value, especially for x values that sit farther from the center.
Use it to check homework, derive Maclaurin expansions, approximate transcendental functions, or build intuition for how polynomials can stand in for curves near a point.
Frequently asked questions
What is a power series?+
A power series is an infinite sum of terms of the form c_k (x - a)^k, where the c_k are constant coefficients and a is the center. The Taylor series is the most common power series: its coefficients are c_k = f^(k)(a) / k!, built from the derivatives of a function f evaluated at the center a.
What is the difference between a Taylor series and a Maclaurin series?+
A Maclaurin series is simply a Taylor series centered at a = 0. So if you set the expansion point to 0 in this calculator, you get the Maclaurin series. Any other value of a gives the general Taylor series about that point.
How do you find the Taylor series of a function?+
Compute the function and its successive derivatives, evaluate each at the center a, divide the k-th derivative by k! to get the coefficient, then multiply by (x - a)^k and add the terms together. This tool automates every step: it differentiates symbolically and assembles the polynomial for you.
How many terms should I use?+
More terms give a closer approximation near the center but can be unwieldy. For a quick estimate, 4 to 6 terms is usually enough; for high accuracy or values farther from the center, use more. This calculator allows up to 20 terms and shows the error against the exact value when you supply an x.
Why is my series inaccurate far from the center?+
A Taylor polynomial matches the function and its derivatives exactly at the center a, so accuracy is best near a and degrades as you move away — and only within the radius of convergence. Functions like 1/(1-x) or log(x) converge only in a limited interval, so evaluating outside it gives large errors.