Taylor Series Calculator

Find the Taylor or Maclaurin series expansion of a mathematical function.

Enter Function f(x): Point (a): Order (n):

🧮 Taylor Series Calculator – Online Function Expansion Tool

Welcome to CalcIreland’s Taylor Series Calculator, your free online tool to calculate and visualize the Taylor and Maclaurin series expansions of mathematical functions with accuracy and ease.
This calculator is designed for students, teachers, researchers, and math enthusiasts in Ireland who want to explore calculus, limits, and series approximation concepts interactively.

For more educational and scientific tools, visit our 👉 Math & Science Calculators section — your hub for free and accurate online calculators designed for students and professionals in Ireland.

🔍 What is the Taylor Series?

In mathematics, the Taylor Series is a powerful tool that represents a function as an infinite sum of its derivatives evaluated at a single point.
It allows us to approximate complicated functions like sin(x), cos(x), exp(x), or ln(1+x) using simple polynomial expressions.

The Maclaurin Series is a special case of the Taylor Series when the expansion point a = 0.

Mathematically, the Taylor Series formula for a function f(x) about a point a is:

$\frac{f”(a)}{2!}(x-a)^2 + \frac{f”'(a)}{3!}(x-a)^3 + \dots$

You can also go through - Radius of Convergence Calculator

⚙️ How to Use the Taylor Series Calculator

Enter your function – Type any valid mathematical expression such as sin(x), e^x, ln(1+x), or x^3 + 2x^2.
Set the expansion point (a) – Choose the point around which you want to expand the function (default is a = 0 for Maclaurin Series).
Select the order (n) – Decide how many terms (order) you want in your Taylor Series expansion (for example, n = 5 for a 5th-order expansion).
Click “Calculate Series” – The calculator instantly computes the Taylor polynomial and displays the result.

You’ll get the simplified polynomial form showing how the function behaves near the chosen point.

🧠 How the Taylor Series Calculator Works

This calculator uses the Taylor series expansion formula and symbolic differentiation to calculate each term up to the desired order.
Here’s what happens behind the scenes:

It takes the input function and automatically computes successive derivatives using the Math.js engine.
Each derivative is evaluated at the given point a.
The result is combined into the Taylor polynomial form:

$Pn(x)=∑k=0nf(k)(a)k!(x−a)kP_n(x) = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!} (x-a)^k$

This gives you an accurate approximation of f(x) near x = a.

🎯 Why Use This Taylor Series Calculator?

✅ Free and Easy to Use – No signup or installation required.
✅ Ireland-specific academic resource – Ideal for Irish secondary schools, colleges, and university students studying calculus or mathematical analysis.
✅ Accurate & Fast – Powered by a modern math engine for instant symbolic computation.
✅ Supports any function – Works for trigonometric, logarithmic, exponential, and polynomial functions.
✅ Learning Tool – Helps students understand how a function can be represented as a polynomial approximation near a point.
✅ Versatile Use – Great for calculus homework, scientific computation, engineering, and research applications.

📚 Example

Example 1:
Find the 4th-order Taylor expansion of f(x) = eˣ around a = 0.

$\frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!}$

That’s exactly what the calculator displays — a simple polynomial approximation of eˣ near zero (Maclaurin Series).

💡 When to Use the Taylor Series Calculator

Use this tool when you need to:

Approximate complex functions with polynomials
Study limits, derivatives, or local behavior of functions
Solve calculus assignments quickly
Verify manual series expansion steps
Analyze engineering or physics functions near a specific value

❓ Frequently Asked Questions (FAQ)

Q1. What is a Taylor Series used for?

A Taylor Series helps approximate complicated mathematical functions with simpler polynomial expressions. It’s widely used in calculus, physics, computer science, and engineering.

Q2. What is the difference between Taylor Series and Maclaurin Series?

The Maclaurin Series is a special case of the Taylor Series where the expansion point a = 0.For example, expanding sin(x) around 0 gives its Maclaurin Series.

Q3. How many terms should I choose in the Taylor Series?

More terms (higher order) give a more accurate approximation. For most practical purposes, 5–7 terms are enough.

Q4. Does the calculator support trigonometric and logarithmic functions?

Yes! You can enter functions like sin(x), cos(x), tan(x), e^x, or ln(1+x) and get their Taylor or Maclaurin expansions instantly.

Q5. Can I use this Taylor Series Calculator on mobile?

Absolutely. The tool is responsive and works smoothly on mobile, tablet, and desktop devices.