Taylor and Maclaurin Series Calculator

Generate common Maclaurin series terms for e^x, sin(x), cos(x), ln(1+x), and 1/(1-x), with convergence reminders.

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Function

Highest term/order

Please enter valid values.

What this calculator teaches

A Maclaurin series is a Taylor series centered at 0. It rewrites a function as a polynomial or power series.

Series are used to approximate functions, solve differential equations, and understand numerical methods.

Formula & Symbols

Concept	Formula	Meaning
Taylor series	`Σ f⁽ⁿ⁾(a)(x-a)^n/n!`	Expansion centered at x=a.
Maclaurin series	`Σ f⁽ⁿ⁾(0)x^n/n!`	Taylor series centered at 0.
Approximation	`f(x) ≈ polynomial`	More terms usually improve accuracy near the center.

Step-by-Step Examples

Example 1

e^x expansion

Use e^x = 1 + x + x²/2! + x³/3! + ...
To order 4, keep powers through x⁴.

✓ 1 + x + x²/2 + x³/6 + x⁴/24.

Example 2

sin(x) expansion

sin(x) uses odd powers only.
Signs alternate: x - x³/3! + x⁵/5! - ...

✓ x - x³/6 + x⁵/120 - ...

Where students use this

🎓

High school and college

Use the tool to check homework and understand each step.

📊

Data and modeling

Connect formulas to tables, graphs, and real values.

🧪

Science and engineering

Use math results inside physics, chemistry, and engineering problems.

💻

Computer science

Apply the same logic to algorithms, systems, and numerical work.

Common Mistakes to Avoid

⚠️

Using far from the center

Maclaurin approximations are most accurate near x = 0.

⚠️

Ignoring convergence

ln(1+x) and 1/(1-x) have interval restrictions.

⚠️

Dropping factorials

Many common series terms use factorial denominators.

Frequently Asked Questions

What is the difference between Taylor and Maclaurin series? ▾

A Maclaurin series is a Taylor series centered at 0.

Why use series? ▾

Series approximate difficult functions using polynomials, which are easier to compute.

Does every function have a useful Taylor series? ▾

Not always. Some functions do not equal their Taylor series over useful intervals.

What does order mean? ▾

Order is the highest power of x kept in the approximation.

Related Math Calculators

Formula Explorer connections

Interpretation: This formula measures limiting behavior, instantaneous change, accumulation or multivariable variation. Assumption: The function must satisfy the continuity or differentiability conditions required by the method. Check domains, bounds, orientation and singularities.

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