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Calculus

Maclaurin Series Expansion

Calculate the Maclaurin series expansion of common functions. Get instant step-by-step solutions — completely free, no signup required.

Free Maclaurin Series Expansion Calculator

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About Maclaurin Series Expansion

Calculate the Maclaurin series expansion of common functions. Calculus is the mathematical study of continuous change. It provides tools for analyzing functions, computing areas and volumes, and solving differential equations that model real-world phenomena.

Our Maclaurin Series Expansion calculator handles every type of problem in this topic. Whether you are a student working through homework, preparing for an exam, or a professional needing a quick calculation, you can enter any maclaurin series expansion expression and receive a complete, verified solution with every step shown.

Maclaurin Series Expansion Formula

f(x) = f(0) + f′(0)x + f″(0)x²/2! + f‴(0)x³/3! + …

A Maclaurin series is a Taylor series centred at a = 0. A handful of standard expansions — eˣ, sin x, cos x, ln(1 + x) and 1/(1 − x) — cover most problems, and new series can be built from them by substitution, multiplication or differentiation.

Worked Example

Find the Maclaurin series of eˣ

  1. Every derivative of eˣ is eˣ, and e⁰ = 1
  2. So every coefficient is 1/n!
  3. Write out the terms: 1 + x + x²/2! + x³/3! + …
  4. Substituting 2x gives e²ˣ = 1 + 2x + 2x² + (4/3)x³ + …

Answer: eˣ = Σ xⁿ / n!

Common Mistakes to Avoid

  • Mixing up the sign patterns of the sin x and cos x series.
  • Forgetting the radius of convergence (e.g. ln(1 + x) only for −1 < x ≤ 1).
  • Substituting into a series but not raising the coefficient as well (e.g. (2x)² = 4x²).

How to Use the Maclaurin Series Expansion Calculator

  1. 1

    Enter your calculus expression (e.g., a function for Maclaurin Series Expansion) into the input field

  2. 2

    Press Enter or click the Solve button to run the calculation

  3. 3

    Review the step-by-step solution that appears below the input

  4. 4

    Check the breakdown of each operation to understand the method

  5. 5

    Try a different problem to practice and build your understanding

Frequently Asked Questions

What is the Maclaurin Series Expansion formula?

f(x) = f(0) + f′(0)x + f″(0)x²/2! + f‴(0)x³/3! + …. A Maclaurin series is a Taylor series centred at a = 0. A handful of standard expansions — eˣ, sin x, cos x, ln(1 + x) and 1/(1 − x) — cover most problems, and new series can be built from them by substitution, multiplication or differentiation.

Can you show a worked Maclaurin Series Expansion example?

Find the Maclaurin series of eˣ: Every derivative of eˣ is eˣ, and e⁰ = 1; So every coefficient is 1/n!; Write out the terms: 1 + x + x²/2! + x³/3! + …; Substituting 2x gives e²ˣ = 1 + 2x + 2x² + (4/3)x³ + …. Answer: eˣ = Σ xⁿ / n!.

Is this Maclaurin Series Expansion calculator free to use?

Yes — the Maclaurin Series Expansion calculator on MathSimplify is completely free. There is no signup, no subscription, and no usage limit. Enter any problem and get the step-by-step solution instantly.

Can I use this for Maclaurin Series Expansion homework and exam practice?

Absolutely. The step-by-step breakdown is designed specifically for learning. You can use it to check your work, understand where you went wrong, or study the method before an exam. The calculator shows the full working, not just the final answer.

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