๐Taylor Series Calculator
Expand a function into a Taylor (or Maclaurin, at a=0) series with term-by-term working and a comparison graph.
Calculations run entirely in your browser using math.js for parsing, symbolic differentiation, and evaluation. No data is sent to a server.
Free Taylor Series Calculator With Steps
Need a polynomial approximation of a function, or want to see how Taylor series actually build up term by term? Our free Taylor series calculator expands any function around a center point instantly. Enter your function, choose the center (0 gives you the Maclaurin series) and the order, and see each derivative term computed and added to the series one at a time.
For common functions like e^x, sin x, and 1/(1-x), it also recognizes the general term pattern and shows the summation notation, connecting the worked-out terms to the formula you'd see in a textbook. A live graph overlays your function with the polynomial approximation, so you can see visually how accuracy improves with more terms and where it starts to diverge.
Enter your function to get started. No sign-up, no downloads, no limits โ just clear, worked-out calculus whenever you need it. See the About page or the FAQ for more.
About the Taylor Series Calculator
Features
Center & order selection
Any center point a and order up to 10.
Term-by-term derivatives
Each derivative order shown as it's computed and evaluated.
General term recognition
Standard functions like eหฃ, sin x, and 1/(1โx) show summation notation.
Comparison graph
Original function overlaid with the Taylor polynomial.
Frequently Asked Questions
How does the Taylor series calculator work?
Enter a function, a center point, and the order, and it computes each derivative term and builds the series step by step.
Is it free?
Completely free, no sign-up, downloads, or limits.
What's the difference between a Taylor and Maclaurin series?
A Maclaurin series is a Taylor series centered at 0; leave the center at 0 to get it.
Does it show the general term formula?
For common functions it recognizes (e.g. e^x, sin x), yes โ alongside the worked-out terms.
How many terms can I expand to?
Up to order 10, which keeps both computation and the step list readable.
Does it graph the approximation?
Yes, it overlays the original function and the Taylor polynomial so you can see how closely they match.
What if the function isn't differentiable at the center?
The calculator detects this and explains why a series can't be built there.
Does it tell me the radius of convergence?
For recognized standard series, yes; otherwise it's omitted rather than guessed.
Is my input stored anywhere?
No; everything runs in your browser and isn't stored or sent anywhere.
Does it work on mobile?
Yes, fully responsive, including the graph and step-by-step panel.