Taylor and Maclaurin Series Calculator (Free)

Use this free Taylor and Maclaurin series calculator to expand a function about any center to a chosen order. It returns the coefficients, the Taylor (or Maclaurin) polynomial, and a graph comparing the approximation to the true function.

Taylor vs. Maclaurin

A Taylor series expands a function around a center a using its derivatives there. When the center is a = 0 it is called a Maclaurin series.

How to use it

1. Pick a function (eˣ, sin x, cos x, ln x, 1/x, 1/(1−x)).
2. Enter the center a and the order n.
3. Press Expand to see the polynomial, coefficients, and graph.

More tools: free math calculators.

Frequently asked questions

What is a Taylor series?

A Taylor series writes a function as an infinite sum of powers of (x − a), built from the function’s derivatives at the center a: f(x) = f(a) + f′(a)(x − a) + f″(a)(x − a)²/2! + … For eˣ about a = 0, every derivative equals 1, so eˣ = 1 + x + x²/2! + x³/3! + …

What is the formula for a Taylor series?

The Taylor series of f about x = a is the sum of f⁽ⁿ⁾(a)(x − a)ⁿ/n! for n = 0, 1, 2 and so on. Each term uses the nth derivative at the center divided by n factorial. For ln x about a = 1, the derivatives give ln x = (x − 1) − (x − 1)²/2 + (x − 1)³/3 − …

What is the difference between Taylor and Maclaurin series?

A Maclaurin series is a Taylor series centered at a = 0, so it uses powers of x instead of powers of (x − a). For example, the Maclaurin series for sin x is x − x³/3! + x⁵/5! − …, while a Taylor series for sin x centered at π uses powers of (x − π). Pick a center close to the x values you want to approximate.

Is Taylor series Calc 2 or Calc BC?

Taylor series are usually taught in Calculus 2 in college, near the end of the unit on sequences and series. In AP Calculus, they are a BC-only topic and are not part of the AB course. You will want to be comfortable with derivatives, factorials and power series before starting them.

How do you calculate Taylor series error?

Use the Lagrange error bound: the error of an nth-degree Taylor polynomial is at most M|x − a|ⁿ⁺¹/(n + 1)!, where M is the largest value of |f⁽ⁿ⁺¹⁾| between a and x. For eˣ ≈ 1 + x + x²/2 at x = 0.1, M is less than 1.2, so the error is below 1.2 × 0.001/6 = 0.0002. The actual error is about 0.00017.

Is a Taylor series a power series?

Yes. A Taylor series is a power series whose coefficients are f⁽ⁿ⁾(a)/n!, taken from a function’s derivatives. In fact, a power series that converges on an interval is the Taylor series of the function it sums to there. For example, 1 + x + x² + x³ + … is both a geometric series and the Maclaurin series of 1/(1 − x), valid for |x| < 1.

How to use the Taylor and Maclaurin Series Calculator for homework

The Taylor and Maclaurin Series Calculator is most useful when you treat it as a learning check, not just a shortcut to the final answer. Start by copying the original problem carefully, including signs, exponents, decimal points, fractions, parentheses, and units. Then enter the values in the same order the problem gives them. A small typing change can completely change the result, especially in algebra, statistics, geometry, and probability problems.

Before you press the button to calculate, make a quick estimate or prediction. The estimate does not need to be exact. Its job is to help you notice impossible answers. If a distance becomes negative, a probability is bigger than 1, an angle looks too large, or a decimal point seems misplaced, go back and check the input before trusting the final result.

Before you enter the problem

• Rewrite the problem in a clean line so every value is easy to see.
• Use parentheses around grouped expressions, especially in fractions and exponents.
• Keep units with the numbers while you work, even if the calculator only asks for the numbers.
• Check whether the problem wants an exact value, a decimal approximation, or a rounded answer.
• Look for restrictions such as positive values only, a chosen interval, or a required domain.

How to read the result

After the calculator gives a result, read more than the final line. If steps, tables, graphs, or intermediate values are shown, use them to understand how the answer was built. That is especially important when you are studying for a quiz or test, because teachers often give more credit for a correct process than for an unsupported number.

Try to identify the main idea behind the result. For example, ask yourself which formula was used, which operation changed the expression, which value controlled the graph, or which assumption made the answer possible. When you can explain that idea in your own words, the tool has helped you learn the skill instead of only checking one problem.

Common mistakes to avoid

Turn the answer into practice

One good way to study is to solve the problem by hand first, then use the Taylor and Maclaurin Series Calculator to check your work. If your answer is wrong, do not erase everything immediately. Find the first line where your work stops matching the calculator's logic. That line is usually where the real misunderstanding happened.

1. Work the problem on paper and circle your final answer.
2. Use the calculator to check the result.
3. If the answers match, write one sentence explaining the method.
4. If they do not match, compare each step until you find the first difference.
5. Redo a similar problem without the tool to make sure the correction sticks.

When to use a calculator and when to work by hand

Use the Taylor and Maclaurin Series Calculator when you want to check a long calculation, explore a pattern, test a graph, or confirm a result after practicing. Work by hand when the assignment asks for steps, when you are learning a new method, or when the test will not allow a digital tool. The strongest students use both: hand work to build understanding and calculators to check accuracy.

If you are preparing for a timed test, practice some problems without the tool and some with it. That balance helps you build speed while still understanding the math. Over time, the goal is to need the calculator less often for routine steps and use it more strategically for checking, exploring, and confirming your reasoning.

Quick accuracy checklist

Before you leave the page, use the Taylor and Maclaurin Series Calculator result to answer three quick questions: What was the input? What rule, formula, or method connects the input to the result? What would change if one number in the problem changed? These questions turn a single calculator answer into a reusable math skill.

For homework, write the original problem, the calculator's result, and one short explanation of the method in your notebook. For test review, cover the result and try to reproduce the setup on your own. If you can set up a similar problem without looking back at the calculator, you are much closer to owning the skill.