T(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)²/2! + ...
f(a) = function value at expansion point
f⁽ⁿ⁾(a) = nth derivative at point a
n! = factorial of n
a = 0 → Maclaurin series
Supported operations:
- Basic: +, -, *, /, ^ (power)
- Trig: sin, cos, tan
- Exp/Log: exp, ln, log
- Other: sqrt, abs, pi, e
Examples:
- sin(x), cos(x), tan(x)
- exp(x), e^x, ln(x)
- x^2 + 3*x - 1
- 1/(1-x), sqrt(1+x)
A Taylor series is a representation of a function as an infinite sum of terms calculated from the values of the function's derivatives at a single point. Named after British mathematician Brook Taylor, this powerful tool allows us to approximate complex functions using polynomials, making calculations and analysis significantly easier. When the expansion point is zero, the series is called a Maclaurin series.
Taylor series are fundamental in calculus, physics, and engineering for approximating functions, solving differential equations, and understanding function behavior near specific points. The more terms included in the series, the more accurate the approximation becomes within the radius of convergence.
Taylor series have numerous practical applications across mathematics and science. In numerical computing, they're used to calculate values of trigonometric, exponential, and logarithmic functions. Scientific calculators and computers use Taylor polynomial approximations for these calculations.
In physics, Taylor series help linearize complex equations for easier analysis, approximate solutions to differential equations, and study small oscillations and perturbations. Engineers use them for signal processing, control systems, and error analysis in measurements.
Taylor series calculations follow standard calculus formulas. Results depend on correct function input, expansion point, and number of terms. This calculator uses numerical differentiation for general functions, which may introduce small approximation errors. For educational and reference purposes only.