What is a Taylor series?

A Taylor series is an infinite polynomial expansion of a function about a centre point a. The series is f(a) + f'(a)(x-a) + f''(a)(x-a) squared divided by 2! + f'''(a)(x-a) cubed divided by 3! and so on. A Maclaurin series is a Taylor series centred at a = 0. By taking more terms you can approximate the function to any desired accuracy over a given interval.

What are the first five terms of the Taylor series for sin(x)?

The Maclaurin series for sin(x) (centred at zero) is x minus x cubed divided by 6 plus x to the fifth divided by 120 minus x to the seventh divided by 5040 plus x to the ninth divided by 362880, and so on. The general term is (minus 1) to the power n times x to the power (2n+1) divided by (2n+1) factorial.

How accurate is a Taylor series approximation?

Accuracy depends on how many terms you include and how far x is from the centre a. For sin(x) at x = 0.5 radians, five terms give an approximation of 0.47943, matching the true value to five decimal places. Adding more terms extends the range of good accuracy further from the centre.

Taylor Series Calculator

A Taylor series is a way of expressing a smooth function as an infinite sum of polynomial terms, each involving a power of (x minus a) where a is the centre point. Truncating the series after a finite number of terms gives a polynomial approximation that can be as accurate as you need, at least near the centre. When the centre is zero the series is called a Maclaurin series. These series are fundamental in calculus, numerical analysis, physics and engineering: they underpin how computers calculate sine, cosine and exponential functions, how engineers approximate complex solutions, and how physicists derive perturbation expansions. The four classic Maclaurin series are for sin(x), cos(x), exp(x) and ln(1+x). This calculator lets you pick one of these functions, choose how many terms to include (up to 10), and enter an x value at which to evaluate the approximation. It returns the polynomial approximation, the exact function value, and the approximation error. It also shows a plain-text representation of the series terms so you can see how the expansion is built up. For sin(0.5 radians), five terms give 0.4794, matching the exact value to four decimal places. The more terms you include, the better the approximation, though for ln(1+x) convergence is only guaranteed for -1 less than x less than or equal to 1.

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Function

Number of terms

x value

0.4794

Taylor series approximation

Exact value0.4794

Approximation error0.0000

Terms used5

x - x^3/6 + x^5/120

x is in radians for sin and cos. For ln(1+x) convergence requires -1 < x ≤ 1.

How it works

The calculator uses the known closed-form Maclaurin series for each function:
sin(x) = x - x³/3! + x⁵/5! - x⁷/7! + ...
cos(x) = 1 - x²/2! + x⁴/4! - x⁶/6! + ...
exp(x) = 1 + x + x²/2! + x³/3! + x⁴/4! + ...
ln(1+x) = x - x²/2 + x³/3 - x⁴/4 + ...
Terms are summed in order. The approximation error is the absolute difference between the series result and the exact value from JavaScript's Math functions.

Worked example

Function: sin(x), 5 terms, x = 0.5 radians. The five terms are: 0.5 - 0.02083 + 0.00026 - 0.0000002 + ... summing to 0.4794. The exact value sin(0.5) = 0.4794 to four decimal places. The approximation error is 0.0000, confirming that five terms are more than enough for this x value.

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