Taylor Series Calculator
Calculate taylor series instantly with our math tool. Shows detailed work, formulas used, and multiple solution methods.
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!\nAt x = 1:\nTerm 0: 1/0! = 1\nTerm 1: 1/1! = 1\nTerm 2: 1/2! = 0.5\nTerm 3: 1/3! = 0.16667\nTerm 4: 1/4! = 0.04167\nTerm 5: 1/5! = 0.00833\nSum = 2.71667\nExact e = 2.71828...\nError = 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!\nAt x = 0.7854 (pi/4):\nTerm 1: 0.7854\nTerm 3: -0.0807\nTerm 5: 0.00249\nTerm 7: -0.0000370\nTerm 9: 0.000000324\nSum = 0.70711\nExact = 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 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.