Taylor Series Calculator
Taylor Series Examples
Select a function and parameters, then click Calculate to see the Taylor series expansion.
sin(x) around x = 0:
x - x³/3! + x⁵/5! - x⁷/7! + ...
e^x around x = 0:
1 + x + x²/2! + x³/3! + x⁴/4! + ...
ln(1+x) around x = 0:
x - x²/2 + x³/3 - x⁴/4 + ...
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. It is used to approximate functions with polynomials, which can be particularly useful for calculations and analysis.
How to Use This Calculator
Enter the function you want to expand, the point around which to expand (usually 0 for Maclaurin series), and the number of terms. The calculator will generate the Taylor series expansion and provide a step-by-step solution.
Formula and Method
The Taylor series of a function f(x) around the point a is given by:
f(x) = f(a) + f′(a)(x-a)/1! + f″(a)(x-a)²/2! + f‴(a)(x-a)³/3! + ...
Applications of Taylor Series
- Approximating complex functions with polynomials
- Evaluating limits and indefinite forms
- Solving differential equations
- Numerical analysis and computational methods
- Physics and engineering calculations
Common Taylor Series Examples
e^x around x=0:
e^x = 1 + x + x²/2! + x³/3! + ...
sin(x) around x=0:
sin(x) = x - x³/3! + x⁵/5! - ...
cos(x) around x=0:
cos(x) = 1 - x²/2! + x⁴/4! - ...
ln(1+x) around x=0:
ln(1+x) = x - x²/2 + x³/3 - ...