Taylor Series Calculator

Enter a function of x, a center a, and an order n to build its Taylor polynomial. Set a = 0 for a Maclaurin series.

f(x)

Center a

Order n

Series —

Frequently Asked Questions

What is a Taylor series?

A Taylor series represents a function as an infinite sum of terms built from its derivatives at a single point a, with each term equal to fⁿ(a)/n! times (x − a)ⁿ. Truncating the sum at a finite order n gives the Taylor polynomial, a local polynomial approximation of the function near a. Keeping more terms makes the polynomial track the original function more closely around that point.

What is the difference between a Taylor and a Maclaurin series?

A Maclaurin series is simply a Taylor series centered at a = 0; every Maclaurin series is a Taylor series, but not every Taylor series is centered at zero. Centering at 0 often simplifies the derivatives, which is why eˣ, sin x and cos x are usually quoted in Maclaurin form. Centering at a nonzero a instead gives a better local approximation when the function is evaluated far from zero.

Which functions can this expand?

This calculator differentiates polynomials, eˣ, sin, cos, ln, and powers or roots of x, then evaluates each derivative at the chosen center to build the series. Combinations of these built with addition, subtraction, multiplication and division are also supported. A function built from a rule this tool does not recognize reports an error rather than a wrong series.

Is my data sent anywhere?

Your figures stay on your device when you use the Taylor series calculator. The Taylor series calculator works in the browser using JavaScript, with no server round-trip at any point, so your function and settings are never transmitted. Nothing is logged, stored or shared with a third party. That also means the tool keeps working offline.

Taylor Series Calculator

Frequently Asked Questions

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