Taylor Series: Approximating Functions with Polynomials

Taylor series are one of mathematics' most elegant tools. They let you express complicated functions as infinite sums of simple polynomial terms. This means you can approximate functions like sine, cosine, and the exponential using nothing more than addition, multiplication, and exponentiation.

The idea is straightforward. If you know the value of a function and all its derivatives at a single point, you can reconstruct the function in a neighborhood around that point. The Taylor series for f(x) centered at a equals f(a) plus f'(a)(x-a) plus f''(a)(x-a)^2/2! plus f'''(a)(x-a)^3/3! and so on.

When centered at zero, these are called Maclaurin series. Some of the most useful Maclaurin series include: e^x equals 1 plus x plus x^2/2! plus x^3/3! plus ...; sine of x equals x minus x^3/3! plus x^5/5! minus ...; cosine of x equals 1 minus x^2/2! plus x^4/4! minus ...

Each additional term in the series improves the approximation, but only within a certain radius called the interval of convergence. For the exponential, sine, and cosine functions, the interval is all real numbers. For the natural logarithm ln(1+x), the series converges only for negative one less than x less than or equal to one.

The remainder term quantifies the error in a truncated Taylor polynomial. Lagrange's form states that the remainder after n terms equals f^(n+1)(c)(x-a)^(n+1)/(n+1)!, where c is some value between a and x. This gives a bound on the approximation error and tells you how many terms you need for a desired accuracy.

Taylor series have practical applications everywhere. Calculators use them to compute trigonometric values. Physicists use them to simplify equations when variables are small. In optimization, Newton's method is essentially a first-order Taylor approximation. In statistics, the delta method uses Taylor expansions to find the variance of transformed random variables.

Our Taylor series calculator generates expansions for common functions. You can compare the truncated polynomial against the exact function to see how quickly the approximation converges. Try the Maclaurin series reference tool for a quick lookup of standard expansions.

One beautiful result connects Taylor series across different functions: Euler's formula, e^(ix) equals cosine x plus i sine x, follows directly from adding the Maclaurin series for cosine and i times sine and comparing with the series for the exponential. This single equation connects five fundamental constants: 0, 1, pi, e, and i.