Polynomials are the easiest functions to work with. You can add them, multiply them, differentiate them, and integrate them using straightforward rules. Most other functions, like sine, cosine, exponential, and logarithm, are harder to manipulate directly. Taylor series bridge this gap by expressing these complex functions as infinite sums of polynomial terms.



The idea is elegant: if you know the value of a function and all its derivatives at a single point, you can reconstruct the entire function (within its radius of convergence) as a power series. For a function f(x) expanded around the point a, the Taylor series is f(a) + f’(a)(x-a) + f’’(a)(x-a)^2/2! + f’’’(a)(x-a)^3/3! and so on. Each term adds more information about the function’s behavior near a.



When a = 0, the series is called a Maclaurin series. Some Maclaurin series are so fundamental that every math student should memorize them. e^x equals 1 + x + x^2/2! + x^3/3! + ... This converges for all x. sin(x) equals x - x^3/3! + x^5/5! - ... Cosine has a similar series with even powers. The natural logarithm ln(1+x) equals x - x^2/2 + x^3/3 - ..., converging for -1 < x <= 1.



Our Taylor series calculator generates these expansions automatically. You can select common functions, choose how many terms to include, and see the coefficients computed step by step. The Maclaurin series reference tool provides a quick lookup for the most important expansions.



The radius of convergence tells you how far from the center the series accurately represents the function. For e^x, sin(x), and cos(x), the radius is infinite, meaning the series works everywhere. For ln(1+x), the radius is 1, and you must check the endpoint x = -1 separately. The ratio test is the standard tool for finding convergence: if lim|a_(n+1)/a_n| = L, then the series converges when L < 1.



Truncating a Taylor series after a finite number of terms gives a polynomial approximation. The error, called the remainder term, depends on the next unused derivative evaluated at some point between your input and the center. For alternating series, the error is bounded by the absolute value of the first omitted term, which is a remarkably useful fact.



In practice, calculators and computers use truncated Taylor series to compute trigonometric and exponential values. When you press the sin button on your calculator, it is essentially evaluating a polynomial that approximates sine to many decimal places. The trigonometric calculator on our site does the same thing using JavaScript’s built-in Math functions.



Taylor series also appear in solving differential equations, analyzing limits, and proving important theorems. They provide a window into the local behavior of functions, and understanding them deeply is essential for advanced work in analysis, physics, and engineering.