Taylor Series Calculator
Calculate taylor series instantly with our math tool. Shows detailed work, formulas used, and multiple solution methods.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Taylor Series Calculator
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Formula: f(x) = sum of f^(n)(a) * (x-a)^n / n! for n = 0 to infinity
Worked example — 6-term approximation = 2.71667 | Exact = 2.71828 | Error = 0.00161
Formula
f(x) = sum of f^(n)(a) * (x-a)^n / n! for n = 0 to infinity
The Taylor series represents a function as an infinite sum of terms involving its derivatives at point a. Each term uses the nth derivative evaluated at a, multiplied by (x-a)^n and divided by n factorial. When a=0, it is called a Maclaurin series.
Worked Examples
Example 1: Taylor Series for e^x at x=1
Problem:Approximate e^1 using the first 6 terms of the Maclaurin series for e^x.
Solution:e^x = 1 + x + x^2/2! + x^3/3! + x^4/4! + x^5/5! At x = 1: Term 0: 1/0! = 1 Term 1: 1/1! = 1 Term 2: 1/2! = 0.5 Term 3: 1/3! = 0.16667 Term 4: 1/4! = 0.04167 Term 5: 1/5! = 0.00833 Sum = 2.71667 Exact e = 2.71828... Error = 0.00161
Result:6-term approximation = 2.71667 | Exact = 2.71828 | Error = 0.00161
Example 2: Taylor Series for sin(x) at x = pi/4
Problem:Approximate sin(pi/4) using 5 terms of the Maclaurin series.
Solution:sin(x) = x - x^3/3! + x^5/5! - x^7/7! + x^9/9! At x = 0.7854 (pi/4): Term 1: 0.7854 Term 3: -0.0807 Term 5: 0.00249 Term 7: -0.0000370 Term 9: 0.000000324 Sum = 0.70711 Exact = 0.70711 (sqrt(2)/2)
Result:5-term approximation = 0.70711 | Exact = 0.70711 | Near-perfect accuracy
Frequently Asked Questions
What is a Taylor series and why is it important?
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. Named after Brook Taylor, it provides a polynomial approximation of functions that can be made arbitrarily accurate by including more terms. The general form is f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + f'''(a)(x-a)^3/3! and so on. Taylor series are foundational in calculus, physics, and engineering because they allow us to approximate complex functions with simpler polynomial expressions. Computers use truncated Taylor series to evaluate transcendental functions like sine, cosine, and exponential.
How many terms do I need for a good approximation?
The number of terms needed depends on the function, the point x where you want to evaluate it, and the center point a of the series. Generally, the closer x is to the center a, the fewer terms you need. For e^x centered at 0, evaluating at x=1 requires about 10 terms for 8 decimal places of accuracy. For sin(x) near x=0, 5-6 terms give excellent accuracy. However, for ln(1+x) with x near 1, convergence is very slow and may require hundreds of terms. The radius of convergence also matters: some series only converge within a certain distance from the center. Outside this radius, adding more terms makes the approximation worse, not better.
What is the radius of convergence?
The radius of convergence R is the distance from the center point within which the Taylor series converges to the actual function value. For |x - a| < R, the series converges; for |x - a| > R, it diverges. At |x - a| = R, convergence depends on the specific series. For example, the Taylor series of e^x has infinite radius of convergence (it works for all x). The series for 1/(1-x) centered at 0 has R = 1 (it only converges for |x| < 1). The series for ln(1+x) at 0 converges for -1 < x <= 1. The radius of convergence is determined by the nearest singularity of the function in the complex plane, even if we only care about real values.
How do computers use Taylor series to calculate functions?
Modern computers use Taylor series (and related polynomial approximations like Chebyshev polynomials) as the core method for evaluating transcendental functions. When you compute sin(x) on a calculator or computer, the hardware or math library first reduces the argument to a small range using identities (e.g., periodicity for trig functions, or scaling for exponentials), then evaluates a carefully optimized polynomial approximation. The coefficients are precomputed for maximum accuracy with minimum terms. Intel x87 floating-point units use polynomial approximations internally. The GNU C library uses Remez-optimized minimax polynomials. These techniques achieve full double-precision accuracy (about 15 decimal digits) with just 6-10 polynomial terms.
What is the Taylor remainder theorem?
The Taylor remainder theorem provides a bound on the error when you truncate the Taylor series after n terms. The Lagrange form of the remainder states that the error equals f^(n+1)(c) * (x-a)^(n+1) / (n+1)! for some c between a and x. This means the error depends on the (n+1)th derivative of the function at some unknown point, the distance from the center raised to the (n+1) power, and the factorial in the denominator. The factorial grows very fast, which is why Taylor series converge: eventually the factorial dominates and each additional term becomes negligibly small. The remainder theorem is essential for determining how many terms are sufficient for a desired accuracy level.
Can every function be represented by a Taylor series?
Not every function has a Taylor series, and having a Taylor series does not guarantee it converges to the function everywhere. A function must be infinitely differentiable at the center point to have a Taylor series. Functions with discontinuities, corners, or vertical asymptotes at the center point cannot have Taylor series there. Even if all derivatives exist, the series might not converge to the function. The classic example is f(x) = e^(-1/x^2) for x not equal to 0 and f(0) = 0: all derivatives at x=0 are zero, so the Taylor series is identically zero, but the function is not zero for x not equal to 0. Functions whose Taylor series converge to themselves are called analytic functions.
How do you find the Taylor series of a product or composition of functions?
For the product of two functions, multiply their Taylor series term by term and collect terms of the same power. For example, to find the series for x * sin(x), multiply x by the series for sin(x). For composition f(g(x)), substitute the series for g(x) into the series for f, then expand and collect terms. This can become algebraically complex. For example, e^(sin(x)) requires substituting the sin series into the exponential series. There are also shortcuts: the Cauchy product formula handles multiplication efficiently, and Faa di Bruno's formula gives derivatives of compositions. In practice, computer algebra systems like Mathematica and SymPy automate these expansions.
What are some applications of Taylor series in physics?
Taylor series are ubiquitous in physics. In mechanics, the small-angle approximation sin(theta) is approximately equal to theta comes from the first term of the Taylor series, enabling analytical solutions for pendulum motion. In electromagnetism, multipole expansions use Taylor series to approximate fields far from charge distributions. In quantum mechanics, perturbation theory is essentially a Taylor expansion of the energy levels and wavefunctions in powers of a small parameter. In thermodynamics, equations of state are often Taylor-expanded around equilibrium. The Born approximation in scattering theory, the post-Newtonian expansion in general relativity, and the virial expansion in statistical mechanics all rely on Taylor series.
What is the difference between Taylor series and Fourier series?
Taylor series approximate functions using powers of (x-a) and work best for functions that are smooth and analytic. Fourier series approximate functions using sines and cosines (periodic basis functions) and can represent discontinuous functions and periodic signals. Taylor series converge locally (near the center point), while Fourier series converge globally over a period. Taylor series require infinitely many derivatives at a point; Fourier series require only integrability. In signal processing, Fourier series decompose signals into frequencies. In numerical analysis, Taylor series provide local polynomial approximations. Both are special cases of more general function space decompositions, and each excels in different applications.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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