Enter a function, a center point, and the number of terms to compute its Taylor (Power) Series expansion. The Power Series Calculator returns the polynomial approximation, individual term coefficients, and a chart showing how each term contributes to the series.
Series Approximation at x
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Exact Function Value at x
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Approximation Error
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Radius of Convergence
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Series Expression (first terms)
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A power series is an infinite sum of the form Σ cₙ(x − a)ⁿ, where x is the variable, a is the center point, and cₙ are the coefficients. It represents a function as a polynomial with infinitely many terms, valid within a certain interval called the radius of convergence.
A Maclaurin series is simply a Taylor series centered at a = 0. Both express a function as a power series using its derivatives; the Maclaurin form is the special case where the expansion point is the origin. Set the center point to 0 in this calculator to get the Maclaurin series.
The radius of convergence R tells you how far from the center point a the series converges. It is typically found using the ratio test or root test on the coefficients. For example, e^x and sin(x) converge for all real x (R = ∞), while 1/(1−x) converges only for |x| < 1 (R = 1).
Not every function can be represented by a power series everywhere. A function must be infinitely differentiable (analytic) at the center point a. Common functions like sin(x), cos(x), e^x, and ln(1+x) have well-known power series, but functions with discontinuities or sharp corners may not.
Classic examples include: e^x = Σ xⁿ/n!, sin(x) = Σ (−1)ⁿ x^(2n+1)/(2n+1)!, cos(x) = Σ (−1)ⁿ x^(2n)/(2n)!, and 1/(1−x) = Σ xⁿ for |x| < 1. These are Maclaurin series (centered at 0) and are used throughout engineering, physics, and numerical analysis.
Accuracy depends on how close x is to the center point and how many terms you include. Near the center, even a few terms give good approximations. Further from the center, or for functions with small radii of convergence, you need more terms. The approximation error shown in the results helps you judge accuracy.
Power series are used in signal processing, physics (quantum mechanics, optics), engineering (control systems, circuit analysis), and numerical computing. They allow computers to evaluate transcendental functions like sin and exp efficiently, and form the basis for techniques like Fourier and Laurent series.
Computing Taylor coefficients for arbitrary symbolic functions requires a full computer algebra system (CAS). This calculator supports the most commonly studied functions with well-known closed-form coefficients, providing exact results. For arbitrary expressions, tools like Wolfram Alpha or eMathHelp offer symbolic computation.