How to use the Taylor and Maclaurin Series Calculator
- 1. Enter f(x)Type the function, for example tan(x), exp(x)*sin(x) or ln(1 + x).
- 2. Choose centre and ordera = 0 gives the Maclaurin series. Choose the highest power n.
- 3. Read the seriesSee f⁽ᵏ⁾(a), each coefficient, the series and its error at your x.
Taylor and Maclaurin series
Near x = a a smooth function can be written as f(x) = f(a) + f′(a)(x − a) + f″(a)(x − a)²/2! + f‴(a)(x − a)³/3! + …. This is the Taylor series about a. With a = 0 it is called the Maclaurin series. Standard ones to remember: eˣ = 1 + x + x²/2! + …, sin x = x − x³/3! + x⁵/5! − …, ln(1 + x) = x − x²/2 + x³/3 − … and tan x = x + x³/3 + 2x⁵/15 + ….
The tool finds the derivatives exactly by series arithmetic, not by numerical differencing, so even the 15th derivative is accurate. Coefficients that are simple fractions are shown as fractions.
Worked example
Expand tan x up to x⁷. At x = 0: f(0) = 0, f′(0) = 1, f″(0) = 0, f‴(0) = 2, f⁽⁵⁾(0) = 16 and f⁽⁷⁾(0) = 272. Dividing by k! gives the coefficients 1, 2/3! = 1/3, 16/5! = 2/15 and 272/7! = 17/315.
So tan x ≈ x + x³/3 + 2x⁵/15 + 17x⁷/315. At x = 0.3 the series gives 0.3093358, while tan 0.3 = 0.3093362, an error of about 4.5 × 10⁻⁷.
Using the series well
A series only represents f inside its radius of convergence; ln(1 + x), for example, needs −1 < x ≤ 1. Keep x close to a for accuracy. Series also give starting steps for ODE methods such as Euler's method.
Frequently asked questions
What is the difference between Taylor and Maclaurin series?
A Maclaurin series is a Taylor series about a = 0. Set the centre to 0 to get it.
How are the derivatives calculated?
By automatic series arithmetic: each operation (product, quotient, sin, exp, ln…) is applied to whole power series. This gives exact derivatives without finite differences.
Why do I get an error for 1/x or ln(x) at a = 0?
These functions are not defined at x = 0, so there is no series about 0. Choose another centre, such as a = 1.
Can I use π in the centre?
Yes. Enter values like pi/4 or pi/6; the series is then written in powers of (x − π/4).
Is this calculator free?
Yes. You fill a short form once when you see your first result, and other Uncodemy tools then open without the form. Everything is calculated in your browser.