Taylor series

The formula

The series itself, expanded about a point a:

f(x) = f(a) + f′(a)·(x−a) + f″(a)/2!·(x−a)² + f‴(a)/3!·(x−a)³ + ...
     = Σ f⁽ⁿ⁾(a)/n! · (x−a)ⁿ    summed over n = 0, 1, 2, ...

Reading: the function's value, slope, curvature, and every higher derivative at one point, each scaled by 1/n!, add up to reconstruct the function nearby. Set a = 0 and it's called a Maclaurin series.

The version engineers actually use — truncate and bound the leftovers:

f(x) = f(a) + f′(a)·(x−a) + ... + f⁽ᵏ⁾(a)/k!·(x−a)ᵏ + Rₖ

Rₖ = f⁽ᵏ⁺¹⁾(ξ)/(k+1)! · (x−a)ᵏ⁺¹    for some ξ between a and x

Reading: the error of a k-term truncation looks like the first term you threw away, with the derivative evaluated somewhere in the interval. That is the Lagrange remainder, and it is the entire theory of truncation error in one line.

The first-order truncation is linearization, the single most used equation in engineering analysis:

f(x) ≈ f(a) + f′(a)·(x−a)

Reading: near the operating point, every smooth system is a straight line. Small-signal models, sensitivity coefficients, and gauge scaling all live here.

The expansions worth having cold (all about zero, x in radians where it matters):

sin x  ≈ x − x³/6          cos x ≈ 1 − x²/2
eˣ     ≈ 1 + x + x²/2      ln(1+x) ≈ x − x²/2        (|x| < 1)
1/(1+x) ≈ 1 − x + x²       √(1+x)  ≈ 1 + x/2 − x²/8  (|x| < 1)

Reading: sin θ ≈ θ is just the series with everything past the first term dropped. The next term tells you what that costs: relative error θ²/6, which crosses 1% at about 14°.

Where you meet it

Bench, small-signal models. A diode's I–V curve is a wall of exponential; the dynamic resistance you use anyway is r_d = V_T/I ≈ 26 mV / I at room temperature — the first derivative of the Taylor expansion about the bias point. Every transistor hybrid-π model, every op-amp linear analysis, is the same move: expand about the operating point, keep the linear term, do circuit theory on the residue.
Test stand, transducer scaling. A quarter-bridge strain gauge's exact output is V_o/V_ex = GF·ε/4 · 1/(1 + GF·ε/2). The datasheet equation drops the second factor — a first-order truncation that costs 0.1% at 1,000 microstrain and grows linearly from there. If your uncertainty budget carries a "bridge nonlinearity" line, that line is a Taylor remainder.
Review board, truncation-error defense. "You linearized the aero model — what's that worth at the corner of the envelope?" The correct answer is a remainder bound, not a shrug. The Lagrange form gives you a defensible worst case: bound the next derivative over the interval, and the error is bounded, in writing, before the board asks.
Flight test, compressibility. The pitot equation's compressible form expands as q_c/q = 1 + M²/4 + ...; at Mach 0.3 the correction is 2.3%, which is exactly why the incompressible airspeed equation is trusted below Mach 0.3 and corrected above it.

How it works

The series trades global knowledge of a function for local knowledge at a point. Each term corrects the previous approximation with the next order of behavior: value, then slope, then curvature. The 1/n! matters — derivatives of well-behaved functions grow slower than n!, so terms shrink fast and a two- or three-term truncation near the expansion point is often good to instrument-grade accuracy. Check: sin 30° from three terms is 0.500002 against the exact 0.5.

Everything hinges on near. The series is a statement about a neighborhood of a, and the neighborhood has a boundary:

Radius of convergence is real, not pedantic. ln(1+x) and √(1+x) converge only for |x| < 1, because the function has a singularity at x = −1 and the series knows it. Feed x = 1.5 into the ln(1+x) expansion and adding terms makes the answer worse. eˣ, sin, and cos converge everywhere — but "converges" and "converges fast enough to truncate" are different claims.
Smooth is not enough. The function e^(−1/x²) (patched to 0 at the origin) has every derivative equal to zero at x = 0, so its Maclaurin series is identically zero — and the function isn't. Infinitely differentiable functions exist whose Taylor series converge to the wrong answer. Physical models rarely bite this way; fitted and piecewise models can.
The error lives in the first dropped term. Practical rule: estimate truncation error by evaluating the next term. For e^0.1 with three terms, the actual error is 1.71×10⁻⁴ and the Lagrange bound is 1.84×10⁻⁴ — the estimate is honest.
Degrees are the classic kill. sin θ ≈ θ requires radians. Run it in degrees and the "approximation" is off by a factor of 57.3. This error still ships in test scripts.
The mistake people make is linearizing once and forgetting they did. The small-angle pendulum model is off by 0.19% in period at a 10° swing — negligible — and 1.7% at 30°, which is not, if you're using the pendulum as a timing or calibration reference. Linearization is a contract with an operating range. Excursions void it, and nothing in the linearized math will warn you; the model stays perfectly self-consistent while drifting away from the hardware. Write the range down next to the model.

History

The polynomial-from-derivatives idea was found at least three times on two continents. The earliest known cases come from Madhava of Sangamagrama (c. 1340–c. 1425) and his Kerala school in southern India, whose series for sine, cosine, and arctangent survive in later texts like Nilakantha's Tantrasangraha (c. 1500) and Jyesthadeva's Yuktibhasa (c. 1530) ^[1]^[6]. Specific series, not the general machine — but two and a half centuries before Europe had either.

In Europe, James Gregory had series for trigonometric functions by early 1671, worked out after correspondence with Newton's circle, though he never published how he got them ^[1]^[2]. Newton himself wrote down an explicit formulation around 1691–92 that stayed in his papers ^[1]^[3]. The general method finally went to print in 1715, in Brook Taylor's Methodus Incrementorum Directa et Inversa, the book that also founded the calculus of finite differences ^[3]^[4]. Taylor (1685–1731) got the theorem into the record and then watched it sit there: its importance went broadly unrecognized until 1772, when Lagrange declared it the main foundation of differential calculus ^[3]^[4]. Lagrange also supplied the piece engineers lean on daily — the explicit remainder term, still standard in the references ^[2]^[7], that turns "approximately" into a bound.

Colin Maclaurin (1698–1746) enters as a naming accident. In his 1742 Treatise of Fluxions he made heavy use of the expansion about zero, credited Taylor for the general result — and got the special case named after him anyway ^[1]^[5].

Related tools

/tools/pendulum-period — the small-angle approximation's most famous customer
/tools/vibration-natural-freq — spring–mass models are restoring forces linearized about equilibrium
/tools/strain-gauge-bridge — quarter-bridge scaling is a first-order truncation with a known remainder
/tools/mach-airspeed — the compressibility correction is the next Taylor term the incompressible equation dropped
/tools/thermal-expansion — L = L₀(1 + αΔT) is a linearization with its operating range printed on it
/tools/rc-charge-time — 1 − e^(−t/RC) is where the eˣ expansion earns its keep at small t

Sources

Written by HE in our own words from the cited sources — engineering judgment included, your stamp still required. All entries →