Taylor Series Calculator
Approximates the value of common functions (eˣ, sin x, cos x, ln x) using their Taylor series expansion around a chosen center point. Ideal for understanding polynomial approximations and truncation error in numerical analysis.
About this calculator
A Taylor series expresses a smooth function 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 0: eˣ ≈ 1 + x + x²/2! + x³/3! + … For sin x: sin x ≈ x − x³/3! + x⁵/5! − … For cos x: cos x ≈ 1 − x²/2! + x⁴/4! − … For ln x centered at a: ln x ≈ Σ [(-1)^(n+1) · (x−a)ⁿ] / (n · aⁿ). Each additional term reduces the approximation error, which is bounded by the remainder term Rₙ = f⁽ⁿ⁺¹⁾(ξ)/(n+1)! · (x−a)^(n+1) for some ξ between a and x. Taylor series underpin numerical computing, signal processing, and physics wherever exact function evaluation is expensive.
How to use
Approximate sin(0.5) using 5 terms centered at a = 0. Set functionType = sin, centerPoint = 0, xValue = 0.5, terms = 5. dx = 0.5 − 0 = 0.5. The series gives: term 1 (n=1): 0.5¹/1! = 0.5; term 2 (n=3): −0.5³/3! = −0.020833; term 3 (n=5): 0.5⁵/5! = 0.000260; term 4 (n=7): −0.5⁷/7! = −0.00000186; term 5 (n=9): negligible. Sum ≈ 0.5 − 0.020833 + 0.000260 − 0.0000019 ≈ 0.47943. The true value of sin(0.5) ≈ 0.47943, confirming excellent accuracy with just 5 terms for x close to the center.
Frequently asked questions
How many terms do I need for an accurate Taylor series approximation?
The required number of terms depends on how far x is from the center point a and how rapidly the function's derivatives grow. For eˣ, sin x, and cos x, whose derivatives are all bounded, convergence is fast — 5 to 10 terms typically give six or more decimal places of accuracy for |x − a| < 1. For ln x, convergence is slower and the series only converges for |x − a| < a. A good rule of thumb is to add terms until successive partial sums differ by less than your desired precision. The calculator's precision field lets you control how many decimal places are displayed.
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, sin x, or cos x around zero, you are computing a Maclaurin series — a special case of the more general Taylor expansion. The general Taylor series is centered at any point a, which is useful when x is far from zero but close to some other convenient reference point. For ln x, centering at a = 1 is common because ln(1) = 0 and the series converges for 0 < x ≤ 2. This calculator supports arbitrary center points for all supported functions.
Why does a Taylor series only approximate a function rather than giving the exact value?
Using a finite number of terms in the Taylor series means you are cutting off an infinite sum, leaving a truncation error called the remainder Rₙ. The full Taylor series, with infinitely many terms, equals the function exactly within its radius of convergence — but computing infinitely many terms is impossible in practice. The Lagrange remainder theorem bounds this error: |Rₙ| ≤ M · |x−a|^(n+1) / (n+1)!, where M is the maximum of |f⁽ⁿ⁺¹⁾| on the interval. This error shrinks as you add more terms or evaluate x closer to the center a, which is why both fields matter for accuracy.