Taylor Series Calculator

About Taylor Series Calculator

This Taylor Series Calculator helps you find the polynomial approximation of any function around a given point. Taylor series are fundamental in calculus and mathematical analysis, providing a way to approximate complex functions with polynomials.

What is a Taylor Series?

A Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. The Taylor polynomial of degree n is the partial sum of the first n+1 terms of the Taylor series.

The general form of 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! + ... + f⁽ⁿ⁾(a)(x-a)ⁿ/n! + ...

How to Use This Calculator

To calculate the Taylor series expansion of a function:

1. Enter the function you want to expand in the "Function f(x)" field (e.g., sin(x), exp(x), log(x))
2. Specify the expansion point (a) around which you want to approximate the function
3. Set the degree (n) of the polynomial approximation
4. Click "Calculate" to see the Taylor polynomial and derivatives

Common Taylor Series Examples

Here are some well-known Taylor series expansions around 0 (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)

Applications of Taylor Series

Taylor series have numerous applications in mathematics and physics:

• Approximating complex functions with polynomials for easier computation
• Solving differential equations
• Evaluating limits (L'Hôpital's rule is often derived using Taylor series)
• Numerical analysis and scientific computing
• Physics and engineering problems where exact solutions are difficult

Important Notes

While Taylor series are powerful tools, there are some limitations:

• The function must be infinitely differentiable at the expansion point
• The series may only converge within a certain radius around the expansion point
• Higher degree polynomials provide better approximations but are more complex
• Some functions (like |x|) don't have Taylor series expansions at certain points