Taylor Series Calculator
Approximate eˣ or sin(x) at any point using a Taylor series expansion centered at a chosen value. Perfect for understanding polynomial approximations in numerical analysis and calculus courses.
About this calculator
A Taylor series expands a function f(x) as an infinite sum of polynomial terms centered at a point a: f(x) = Σ [f⁽ⁿ⁾(a) / n!] · (x − a)ⁿ. For eˣ centered at a = 0 this becomes: eˣ ≈ 1 + x + x²/2! + x³/3! + … For sin(x) centered at a = 0: sin(x) ≈ x − x³/3! + x⁵/5! − x⁷/7! + … Using more terms (higher n) captures more of the function's behavior and reduces approximation error. The error of an n-term approximation is bounded by the (n+1)-th term of the series, so convergence is rapid near the center point a.
How to use
Approximate sin(0.5) using 4 terms, centered at a = 0. Set function = sin, x = 0.5, center = 0, terms = 4. Step 1: Term 1 (n=0): 0.5¹/1! = 0.5. Step 2: Term 2 (n=1): −0.5³/3! = −0.125/6 ≈ −0.02083. Step 3: Term 3 (n=2): 0.5⁵/5! = 0.03125/120 ≈ 0.000260. Step 4: Term 4 (n=3): −0.5⁷/7! ≈ −0.0000016. Sum ≈ 0.47943. The true value of sin(0.5) ≈ 0.47943 — excellent accuracy with just 4 terms.
Frequently asked questions
How does a Taylor series approximation work and why is it useful?
A Taylor series replaces a complex function with a polynomial that matches the function's value and all its derivatives at a chosen center point. Polynomials are easy to evaluate, differentiate, and integrate, making them invaluable in numerical computing, physics simulations, and engineering models. The more terms you include, the wider the interval over which the polynomial closely tracks the original function. This is the foundation of many numerical methods used in scientific software.
What is the difference between a Taylor series and a Maclaurin series?
A Maclaurin series is simply a Taylor series centered at a = 0. When you expand eˣ or sin(x) around zero, you are computing a Maclaurin series. A Taylor series is the more general form that allows any center point a, which is useful when you want to approximate a function near a point other than the origin. Both are otherwise identical in structure and derivation.
How many terms do I need for an accurate Taylor series approximation?
It depends on how far x is from the center point a and how much accuracy you need. For eˣ and sin(x), the series converges for all real x, but convergence is faster when |x − a| is small. In practice, 5–10 terms usually give accuracy to 6 or more decimal places for |x − a| < 1. If x is far from the center, you need significantly more terms or should choose a center closer to x.