Taylor Series Calculator
Expand functions into Taylor and Maclaurin series with step-by-step solutions
Use 0 for Maclaurin series
Compare series approximation with actual function value
Taylor Series Expansion (Maclaurin Series)
centered at a =
Term-by-Term Breakdown
Approximation at x =
Series Approximation
Actual Value
Error
(%)
eˣ
1 + x + x²/2! + x³/3! + x⁴/4! + ...
sin(x)
x − x³/3! + x⁵/5! − x⁷/7! + ...
cos(x)
1 − x²/2! + x⁴/4! − x⁶/6! + ...
ln(1+x)
x − x²/2 + x³/3 − x⁴/4 + ...
1/(1−x)
1 + x + x² + x³ + x⁴ + ...
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About Taylor Series Calculator
What is a Taylor Series?
A Taylor series is a representation of a function as an infinite sum of terms calculated from the function's derivatives at a single point. It allows complex functions to be approximated by polynomials, making computations easier.
The Taylor Series Formula
The Taylor series expansion of a function f(x) around a point 'a' is:
f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)²/2! + f'''(a)(x-a)³/3! + ...
Or using sigma notation:
f(x) = Σ (from n=0 to ∞) [fⁿ(a)/n!] × (x-a)ⁿ
Maclaurin Series
A Maclaurin series is a special case of Taylor series where the expansion is centered at a = 0:
f(x) = f(0) + f'(0)x + f''(0)x²/2! + f'''(0)x³/3! + ...
How to Use This Calculator
- Select Function: Choose from common functions (sin, cos, eˣ, ln, etc.)
- Enter Center Point: Specify the point 'a' around which to expand (0 for Maclaurin)
- Set Terms: Choose how many terms to compute (3 to 10)
- Calculate: View the complete expansion with coefficients
Common Taylor/Maclaurin Series
| Function | Series (centered at 0) |
|---|---|
| eˣ | 1 + x + x²/2! + x³/3! + ... |
| sin(x) | x - x³/3! + x⁵/5! - x⁷/7! + ... |
| cos(x) | 1 - x²/2! + x⁴/4! - x⁶/6! + ... |
| ln(1+x) | x - x²/2 + x³/3 - x⁴/4 + ... |
| 1/(1-x) | 1 + x + x² + x³ + ... |
Applications of Taylor Series
- Approximating functions with polynomials
- Computing limits that are otherwise difficult
- Solving differential equations
- Error estimation in numerical methods
- Physics and engineering calculations
Frequently Asked Questions
What is the difference between Taylor and Maclaurin series?
A Maclaurin series is simply a Taylor series centered at x = 0. Both use the same formula.
How many terms do I need for a good approximation?
The number of terms depends on the function and how far x is from the center point. Generally, more terms give better accuracy.
What functions can be expanded as Taylor series?
Functions that are infinitely differentiable at the center point can be expanded as Taylor series.
Disclaimer: This calculator is for educational purposes. Results are approximations based on the number of terms computed.