The Maclaurin Series Formula

At 2:00 AM, an engineering student needs sin(0.2) for a vibration problem and has no calculator. The Maclaurin series formula gives the answer in three terms: 0.2 minus one six-thousandth plus a tiny correction, matching the true value to five decimal places. This is the power of the Maclaurin series formula: it turns complicated functions into polynomials you can compute by hand.

The Maclaurin series is a Taylor series centered at x = 0. The definition is f(x) = Σ_{n=0}^{∞} (f⁽ⁿ⁾(0)/n!) xⁿ. Here f⁽ⁿ⁾(0) means the n-th derivative evaluated at zero, and n! is the factorial of n. This is what a Maclaurin series is: an infinite polynomial that matches a function's value and all its derivatives at the origin.

What Is a Maclaurin Series

A Maclaurin series represents a function as an infinite sum of power terms, each weighted by a derivative at zero. For eˣ, the series is 1 + x + x²/2! + x³/3! + … . For sin x, it is x, x³/3! + x⁵/5!, … . These series are not approximations, they equal the function exactly for every x within their interval of convergence.

To derive the Maclaurin series, start with a general power series f(x) = c₀ + c₁x + c₂x² + c₃x³ + … . Evaluate at x = 0: f(0) = c₀. Differentiate once: f'(x) = c₁ + 2c₂x + 3c₃x² + … . Evaluate at 0: f'(0) = c₁. Differentiate again: f''(x) = 2c₂ + 6c₃x + … . Evaluate at 0: f''(0) = 2c₂, so c₂ = f''(0)/2. For the third derivative: f'''(0) = 6c₃, so c₃ = f'''(0)/6 = f'''(0)/3!.

This derivation shows why the factorial appears: it comes from the repeated differentiation of xⁿ, which gives n·(n−1)·…·1 = n!. The claim that n! 'ensures the series converges' is misleading. The factorial grows fast, but the ratio test determines convergence. For sin x and eˣ, the factorial overwhelms xⁿ for any finite x, giving infinite radius. For ln(1+x), the geometric terms without factorial give radius 1.

The Maclaurin polynomial Pₙ(x) is the first n+1 terms of the series. For sin x, P₃(x) = x, x³/6 gives 0.198 for x = 0.2, already accurate to 0.3%. The infinite series is the limit as n → ∞. Within the radius of convergence, the series equals the function. Outside it, the series diverges or converges to the wrong number.

The Lagrange remainder Rₙ(x) = f⁽ⁿ⁺¹⁾(c)/(n+1)! xⁿ⁺¹ for some c between 0 and x gives the error of Pₙ(x). Taylor's inequality bounds it: |Rₙ(x)| ≤ M |x|ⁿ⁺¹/(n+1)!, where M bounds |f⁽ⁿ⁺¹⁾| on [0,x]. This bound overestimates the true error by a factor of 10 or more because M is a worst-case value. The alternating series error bound is tighter: for alternating series, the error is ≤ the magnitude of the first omitted term.

A function that equals its Maclaurin series on an open interval around 0 is called analytic. All polynomials, eˣ, sin x, cos x, and rational functions without poles at 0 are analytic. But not every smooth function is analytic. The classic counterexample is f(x) = e^(, 1/x²) for x ≠ 0, with f(0) = 0. All derivatives at 0 are zero, so its Maclaurin series is 0 + 0x + 0x² + … = 0. For any x ≠ 0, the series equals 0 while the function is positive. This function, from Stewart 'Calculus' §11.10, shows that a function can have a valid Maclaurin series that does not represent it.

The ratio test determines the radius of convergence. For eˣ, the limit L = 0, giving radius ∞. For ln(1+x), L = 1, giving radius 1, but the series converges at x = 1 (conditionally) and diverges at x =, 1. Check endpoints separately.

Colin Maclaurin (February 1698-14 January 1746, per MacTutor History of Mathematics) was a Scottish mathematician and a student of Isaac Newton. He used Taylor series centered at 0 so extensively that the special case took his name. Brook Taylor (18 August 1685-29 December 1731) had discovered the general form earlier. Maclaurin did not claim priority; the name stuck because of his applications in geometry and series expansions.

Maclaurin series are for Calculus II and AP Calculus BC students who need to compute series and find intervals of convergence for exams. Engineering students who approximate small-angle pendulum motion or electronic circuits use them with a known error bound. Physics students in perturbation theory replace functions with Maclaurin polynomials and need the convergence radius. Self-learners check their work against a calculator that shows terms and error bounds.

Skip Maclaurin series if you need a Taylor series centered at a nonzero point, like approximating ln x near x = 2. That requires a general Taylor series, not the Maclaurin special case. Also skip if you need a proof of Taylor's theorem with integral remainder, that is real analysis, not calculus.

The single thing that most often goes wrong: forgetting to check endpoints. The ratio test gives the radius but not the interval. A series that converges for |x| < R may converge, diverge, or converge conditionally at x = ±R. Test both endpoints separately. Without that check, your interval of convergence is wrong.