Power series

The formula

f(x) = Σ a_n (x − c)ⁿ  =  a₀ + a₁(x − c) + a₂(x − c)² + a₃(x − c)³ + ...
        n=0..∞

Reading: pick a center c, and approximate the function as an endless polynomial in the offset (x − c). Each coefficient a_n weights one power.

1/R = limsup |a_n|^(1/n)        (Cauchy–Hadamard)

Reading: the radius of convergence R is set by how fast the coefficients grow. Big coefficients that don't shrink mean a small R; coefficients that die off fast mean a large one.

R = lim | a_n / a_(n+1) |       (ratio-test form, when the limit exists)

Reading: the everyday shortcut. Take the ratio of consecutive coefficients; its limit is the radius. Easier to compute by hand than the root form, and it agrees whenever it converges.

Converges for  |x − c| < R
Diverges  for  |x − c| > R
Boundary  |x − c| = R : no general rule — check case by case

Reading: inside the circle the series is a faithful copy of the function. Outside it, the partial sums fly apart. On the rim, anything can happen.

Geometric:  1/(1 − x) = 1 + x + x² + x³ + ...   with R = 1

Reading: the cleanest example. Center c = 0, radius exactly 1. At x = 0.5 the sum is 2; at x = 0.9 it climbs to 10; at x = 1 it refuses to settle.

Where you meet it

Small-angle approximations on a vibration bench. sin θ ≈ θ − θ³/6 and sin θ ≈ θ are the first terms of the sine series. A pendulum-period rig or a control loop linearized about equilibrium leans on that truncation. At 10° amplitude the pendulum period is off by only 0.19%; at 30° it's 1.74% — the higher terms you threw away are exactly that error. (The first correction term alone, θ²/16, predicts 1.71%; the missing 0.03% is the next term in the series, a truncation error inside your truncation-error estimate.)
Filter and transfer-function analysis. Expanding 1/(1 + sRC) or a feedback loop 1/(1 − Gβ) as a geometric series is how you reason about small-signal behavior and loop gain. The expansion is honest only while the offset stays inside R; push past it and your linearized model is quietly lying.
Firmware and DSP math libraries. exp, sin, log, and atan on a microcontroller with no FPU are computed from truncated series (often after range reduction). The number of terms kept is a direct trade between cycles burned and bits of accuracy — a radius-of-convergence question dressed up as a runtime budget.
Review-board sanity checks. When someone presents a curve fit or a "linear region" on a test-report slide, the unasked question is: how far from the fit point does this stay valid? That distance is a radius of convergence in spirit, and it's the first thing a sharp reviewer probes.

How it works

A power series is a polynomial with no last term. Near the center c the low-order terms dominate and a handful of them reproduce the function to whatever accuracy you need. Walk away from c and the neglected high-order terms wake up. Cross the radius R and they win outright — the partial sums diverge no matter how many you keep.

The radius is not a suggestion. It is a hard wall set by the function itself, and here's the part that catches people: the wall can sit at a place where the real function looks perfectly healthy. Take 1/(1 + x²). It's smooth and finite for every real x — no blowup anywhere on the number line. Yet its series about 0 is 1 − x² + x⁴ − x⁶ + ... with radius exactly 1. At x = 0.5 the series gives 0.8, dead-on. Push to x = 1.1 and forty terms already read −926 against a true value of 0.45. The reason lives off the real axis: the function has poles at x = ±i, distance 1 from the origin, and the series can't see past the nearest trouble in the complex plane even when you only care about real inputs. The radius is the distance to the closest singularity, full stop.

The common mistake is using a series-based model outside its radius and trusting the answer because "it's just a polynomial." Truncated series don't warn you gently — they degrade slowly, then diverge violently. A Taylor-fit gain curve that's flawless across its fit window can be off by a factor of a thousand a short step beyond it, with no kink to tip you off on the way out.

Two more field notes. First, more terms buy you accuracy inside R, never a bigger R — the wall doesn't move by working harder. If you need to go farther, re-center the series closer to where you're operating (a fresh c), or switch functions. Second, the boundary itself is genuinely undecided. The geometric series diverges at both x = ±1; the log series x − x²/2 + x³/3 − ... for ln(1 + x) shares the same radius 1 but actually converges at x = 1 and diverges at x = −1. Same radius, opposite fates on the rim. Never assume the endpoints behave — test them.

History

The idea arrived before the rigor did. In the mid-1660s Isaac Newton was expanding functions as infinite series and using them to integrate things nobody could integrate in closed form, but he sat on the work. It was Nicholas Mercator who first put a power series into print, in his 1668 Logarithmotechnia, giving ln(1 + x) = x − x²/2 + x³/3 − ... ^[1]^[2]. James Gregory was independently turning out series expansions in the same stretch of years. The tool worked; nobody yet asked where it stopped working.

Brook Taylor generalized the move in his 1715 Methodus Incrementorum Directa et Inversa, showing how to build a series for a general function from its derivatives at a point — what we now call the Taylor series, though the name didn't attach until decades later ^[3]^[4]. Colin Maclaurin popularized the special case centered at zero, and it carries his name.

For a century the field expanded functions freely and mostly got away with it. Then Augustin-Louis Cauchy, in his 1821 Cours d'analyse, made convergence a question you had to answer rather than assume, and stated the coefficient test that pins down exactly how far a series can be trusted ^[5]. The result went quiet until Jacques Hadamard rederived and published it — first in 1888, then in his 1892 doctoral thesis — which is why the radius formula now carries both their names as the Cauchy–Hadamard theorem ^[5]. Niels Henrik Abel had meanwhile nailed down the delicate case: what happens right on the boundary circle, where the general rules give out. The sequence is worth remembering: engineers used the series for 150 years before anyone proved where the edge was. The edge was always there.

Related tools

/tools/pendulum-period
/tools/rc-filter
/tools/series-rlc-impedance
/tools/vibration-natural-freq
/tools/euler-buckling

Sources

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