The Taylor Series Calculator expands functions into infinite polynomial series, providing powerful approximations of complex functions near a specific point. Taylor series are fundamental in mathematical analysis, numerical methods, physics, and engineering, allowing us to approximate transcendental functions using only polynomial operations.

What is a Taylor Series?

A Taylor series represents a function f(x) as an infinite sum of terms calculated from the function's derivatives at a single point a:

f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)²/2! + f'''(a)(x-a)³/3! + ...

Or more compactly: f(x) = Σ_n₌₀^∞ [fⁿ(a)/n!]·(x-a)ⁿ

Special Cases

• Maclaurin Series: Taylor series centered at a = 0. Many common series are Maclaurin series.
• Power Series: General form Σa_nxⁿ. Taylor series are power series with specific coefficients.
• Polynomial Approximation: Truncating after n terms gives the nth-degree Taylor polynomial, approximating f near a.

Common Taylor/Maclaurin Series

• e^x = 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 + ... (for |x| < 1)
• 1/(1-x) = 1 + x + x² + x³ + ... (for |x| < 1)

Convergence and Radius

A Taylor series may converge for all x, or only for x within a certain radius of convergence R around the center point a. Within the radius of convergence, the series equals the function exactly. Outside this radius, the series may diverge or converge to a different value.

Taylor Series Examples

Tips for Working with Taylor Series

• Memorize Common Series: Know the series for e^x, sin(x), cos(x), 1/(1-x), and ln(1+x) by heart.
• Use Known Series: Instead of computing derivatives, substitute into known series. For e^(-x), use e^x series with -x.
• Check Radius of Convergence: Always determine where the series converges before using it for approximations.
• Even/Odd Functions: Even functions have only even powers, odd functions only odd powers in Maclaurin series.
• Truncation for Approximation: For practical calculations, use enough terms to achieve desired accuracy.
• Alternating Series Error: For alternating series, the error is less than the first omitted term.
• Operations on Series: You can add, subtract, multiply, and differentiate/integrate series term by term (within radius of convergence).
• Substitution Technique: To find series for f(g(x)), substitute the series for g(x) into the series for f(x).
• Compare with Polynomial: The nth Taylor polynomial is the best polynomial approximation of degree ≤ n at the center point.
• Practical Applications: Calculators and computers use Taylor series to compute trigonometric and exponential functions!