Compute Taylor & Maclaurin series expansions instantly with step-by-step solutions, interactive graphs, and convergence analysis
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Visualization shows the original function (blue) and Taylor polynomial approximation (red). Adjust the degree to see how accuracy improves with more terms.
Calculate Taylor series expansions up to 20th degree in milliseconds. Our optimized algorithms compute derivatives and series terms instantly.
Visualize how Taylor polynomials approximate functions. Watch the approximation improve as you increase the degree. Real-time graphing with Plotly.js.
Learn Taylor series concepts with detailed step-by-step solutions showing each derivative calculation. Perfect for students learning calculus.
Get insights about radius of convergence and approximation accuracy. Understand where the Taylor series provides valid approximations.
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A Taylor series is a mathematical representation of a function as an infinite sum of terms calculated from the function's derivatives at a single point. Developed by Brook Taylor in 1715, it's one of the most powerful tools in calculus for approximating complex functions with polynomials.
The general formula for the Taylor series expansion of a function f(x) about point a is:
When the expansion point a = 0, the series is called a Maclaurin series (named after Colin Maclaurin). Taylor series are fundamental in physics, engineering, and computer science for approximating functions that are difficult to compute directly.
Taylor series approximate complex equations in structural analysis, fluid dynamics, and electrical circuit design. Engineers use them to simplify calculations while maintaining accuracy.
Programming languages use Taylor series to compute trigonometric, exponential, and logarithmic functions. Calculators and computers implement these approximations in hardware.
In quantum mechanics, perturbation theory uses Taylor expansions. Classical mechanics employs them for small oscillations and approximations in potential energy functions.
Taylor series approximate utility functions in economics and option pricing models in finance. Risk assessment models rely on these polynomial approximations.
A Maclaurin series is a special case of Taylor series where the expansion point a = 0. So every Maclaurin series is a Taylor series, but not vice versa. Use our calculator with center point 0 for Maclaurin series.
It depends on the function and how far x is from the center point. For functions like eˣ and sin(x), 5-10 terms give excellent accuracy near the center. Use our interactive graph to visualize accuracy.
The radius of convergence is the distance from the center point within which the Taylor series converges to the function. Outside this radius, the series diverges. Our calculator provides convergence information.
Taylor series work for functions that are infinitely differentiable at the expansion point. Functions with discontinuities or singularities may not have valid Taylor series expansions.
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