One point's worth of information

Stand at one point on a curve and measure everything measurable there: the height, the slope, how fast the slope is changing, how fast that is changing, and onward. Every one of those is a local quantity — each is determined by an arbitrarily small neighbourhood of the point, and none of them looks anywhere else.

The claim of Taylor’s theorem is that this list is enough to reconstruct the whole curve.

The construction #

The recipe writes itself once the goal is stated. Build a polynomial that matches the function at a point, then matches its slope, then its second derivative, and so on:

$$f(x) \approx f(0) + f'(0)\,x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots$$

The factorials are not decoration. Differentiating $x^k$ three times gives $k(k-1)(k-2)x^{k-3}$, and by the time it has been differentiated $k$ times the accumulated factor is exactly $k!$. Dividing by $k!$ cancels it, so each coefficient is precisely the derivative it is supposed to reproduce and nothing else — the terms do not interfere with each other’s jobs.

That independence is the whole design. The degree-5 term contributes nothing to the first four derivatives at the origin, because it and its first four derivatives all vanish there. So a partial sum of degree $n$ matches the function’s first $n$ derivatives at the point exactly, and adding another term never disturbs what the earlier ones settled.

For $\sin x$ the derivatives at zero cycle $0, 1, 0, -1$ forever, which kills every even power and alternates the signs of the odd ones:

$$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$$

The first term alone says $\sin x \approx x$, which is the small-angle approximation every physics course uses without justification. It is the degree-1 Taylor polynomial, and its error is the next term — about $x^3/6$, which at $5°$ is one part in three thousand and at $30°$ is four percent. That is where the pendulum’s honest period goes wrong, and now it has a number attached.

Each term buys an interval #

The figure’s real content is the way the curves peel away.

Degree 1 is right near the origin and hopeless by the first bend. Degree 3 survives past the first hump. Degree 9 tracks $\sin x$ through several. The pattern is not that higher degree is more accurate everywhere — it is that higher degree is accurate over a wider interval, and outside that interval it is usually worse than the low-degree version, because a high-degree polynomial that has run out of agreement diverges violently.

$e^x$ shows the mechanism cleanly because nothing cancels. All its derivatives at zero are $1$, so the series is $1 + x + x^2/2 + x^3/6 + \cdots$, every term positive. A partial sum is therefore always an underestimate to the right of zero, and the deficit is the tail — which is the same series scaled, and shrinks factorially. This is the series that computes $e$ to seven places in ten terms, and the reason is visible here: the factorial in the denominator eventually overwhelms any fixed $x$.

The remainder can be written down. Taylor’s theorem with Lagrange’s form of the remainder says that after $n$ terms the error is

$$\frac{f^{(n+1)}(\xi)}{(n+1)!}\,x^{n+1}$$

for some $\xi$ between $0$ and $x$. The location $\xi$ is unknown, which is the catch — but a bound on the derivative is usually available, and that is enough. There is also an exact form, in which the remainder is an integral of the next derivative against a weight, and it is the version to reach for when a bound is not good enough and the derivative can actually be integrated. For $\sin x$ every derivative is bounded by $1$, so the error after $n$ terms is at most $|x|^{n+1}/(n+1)!$, which goes to zero for every $x$ whatever. That is the proof that the sine series works everywhere, and it takes one line once the remainder is in hand.

The function that ends the optimism #

So far the story is that the local data determines the global function, and the sine and the exponential both cooperate. Now the case that does not, and it is not a pathology — it is a rational function that a school student could write down.

$$f(x) = \frac{1}{1 + x^2}$$

This function is perfectly well behaved on the entire real line. It is smooth everywhere. It has derivatives of every order at every point. It never blows up, never has a corner, never does anything at all except drop gently from $1$ toward zero in both directions.

Its Taylor series about zero is

$$1 - x^2 + x^4 - x^6 + \cdots$$

which is a geometric series in $-x^2$, and geometric series converge only when the ratio is less than one in size. So the series converges for $|x| < 1$ and diverges for $|x| > 1$.

At $x = 1.4$ the degree-4 sum is out by about $2$; the degree-28 sum is out by roughly $500$. Adding terms makes it worse, without limit, at a point where the function is entirely unremarkable. Nothing at $x = 1.4$ is wrong. Nothing at $x = \pm 1$ is wrong either — the function sails through both.

The generator checks exactly this, because it is the essay’s central claim and easy to get backwards: it computes the error at a point inside the radius and at a point outside, and requires the first to shrink with degree and the second to grow.

Why one, and why nobody can see it #

The radius is $1$, and there is nothing at distance $1$ along the real line to explain it. The explanation is not on the real line at all.

Extend the function to complex inputs. Then $1 + x^2$ vanishes at $x = i$ and $x = -i$, and $f$ blows up there. Those points are at distance exactly $1$ from the origin.

A Taylor series about a point converges inside the largest disc — in the complex plane — that contains no singularity, and it diverges outside it. The radius of convergence is the distance to the nearest place the function misbehaves, measured in the plane whether or not the function was ever meant to leave the line.

That is a genuinely startling explanation. The behaviour of a real function of a real variable, on the real axis, is governed by two points that are nowhere on the real axis. The real line is a slice through a two-dimensional situation, and the slice cuts through a disc whose radius was set by something off the slice entirely — a situation multiplying as turning is the right language for and which no real picture can carry.

So the failure has nothing to do with smoothness, which is why examining $f$ on the real line will never turn it up. A function can be infinitely differentiable everywhere and still have a radius of convergence of $1$, or $0.001$, or — as the next section shows — nothing at all.

Smooth is not enough, in the worst way #

There is a sharper counterexample, and it kills the hope entirely.

$$g(x) = \begin{cases} e^{-1/x^2} & x \ne 0 \\ 0 & x = 0\end{cases}$$

This function is smooth everywhere, including at zero. Every derivative exists there, and every single one of them is zero — the exponential decays faster than any polynomial grows, so every derivative is dragged to nothing at the origin.

Its Taylor series about zero is therefore

$$0 + 0 \cdot x + 0 \cdot x^2 + \cdots = 0.$$

The series converges beautifully, everywhere, with infinite radius. It converges to the zero function. And $g$ is not the zero function — it is positive at every $x \ne 0$.

So here the series converges and converges to the wrong thing. The local data at the origin — every derivative, all infinitely many of them — is completely blind to what $g$ does anywhere else. The claim this essay opened with is simply false for $g$.

Functions for which the claim is true have a name: analytic. Analytic implies smooth; smooth does not imply analytic, and $g$ is the standing witness. On the real line the two classes are enormously different. In the complex plane they collapse — a complex function differentiable once is automatically analytic, which is one of the more surprising facts in mathematics and is the reason complex analysis is a tidier subject than real analysis.

Nobody computes with these, and everybody uses them #

There is an apparent contradiction worth resolving. Taylor series are the standard way of introducing infinite sums, and essentially no working numerical routine evaluates a function by summing one.

Library implementations of $\sin$, $\exp$ and the rest use minimax polynomials instead — polynomials of the same degree chosen to make the worst error over the interval as small as possible, rather than to be perfect at one point and progressively worse away from it. The difference is substantial. A degree-7 Taylor polynomial for $\sin$ on $[-\pi/4, \pi/4]$ is exact at the origin and worst at the ends; a degree-7 minimax polynomial is slightly wrong everywhere and its worst case is smaller by a factor of tens. Since a library cares only about the worst case, it uses the second, and the Taylor coefficients appear nowhere in the shipped code.

So what is the series for? Almost always, for its first two or three terms, used as a licence rather than as a computation.

To first order is the most-used phrase in applied mathematics, and it means: replace the function by its degree-1 Taylor polynomial and accept an error of order $x^2$. That single move is linearisation, and it is what makes a pendulum a harmonic oscillator, a small perturbation solvable, an error bar propagate, and a nonlinear system tractable near an equilibrium. Newton’s method is the degree-1 sum solved for its root and iterated. Error propagation in an experiment is the degree-1 sum with the derivative playing the role of a sensitivity.

It is worth seeing what linearisation is in the language of the rest of this collection: the degree-1 term is a number times $x$, which is to say a linear map — and in several variables it is literally a matrix, the Jacobian. “To first order” means “replace the function by the linear map that best matches it here”, and everything known about linear maps then becomes available. The reason the move is so productive is not that polynomials are simple; it is that linear algebra is finished and nothing else is.

None of that needs convergence, or a radius, or the infinite tail. It needs the remainder bound — the guarantee that the error is $O(x^2)$ — which Taylor’s theorem supplies with finitely many terms and no infinite series anywhere. The series is the famous object; the theorem with a remainder is the useful one, and they are routinely confused.

What the picture cannot show #

The figures draw partial sums, and a partial sum is a polynomial. Every one of them is smooth, unbounded, and eventually enormous — none of them is the function, and the function is not among them.

More acutely, the pictures cannot show the boundary. The vertical lines at $\pm 1$ in the radius figure are drawn there because the mathematics puts them there, not because anything visible happens at that spot. A reader shown the curve alone could look at $x = 1$ for a very long time without finding a reason for it, because the reason is at $x = \pm i$ and the page has no axis for it.

Nor can any drawing separate smooth from analytic. $g(x) = e^{-1/x^2}$ plotted next to any other gently rising curve looks unremarkable — it is flat near zero and then climbs, which describes a great many functions. That its every derivative vanishes at the origin, and that this makes its Taylor series useless, has no visual signature at all. The failure is a statement about a limit of ratios and it is invisible at every scale.

The ladder from here #

Rungs above: the remainder in Lagrange and integral forms, with the bound drawn. Expanding about a point other than zero, and why the choice matters for accuracy. The radius of convergence proved from the ratio test. Analytic continuation, where a function is extended by re-expanding at a new centre and walking. Complex differentiability and why once implies always. Padé approximants, which beat Taylor by allowing a denominator and reach past the radius. Runge’s phenomenon, where interpolating at evenly spaced points fails on the same $1/(1+x^2)$ for a related reason. Numerical evaluation of elementary functions, where the series is never used raw and the argument reduction is the hard part. And Fourier series, the other way of writing a function as an infinite sum — which uses global information rather than local, and succeeds on functions Taylor cannot touch.

Local against global #

The comparison with Fourier is the most useful thing to leave with, because the two constructions answer the same question from opposite ends.

A Taylor series is built from data at one point and is exact near it. A Fourier series is built from integrals over the whole period and is accurate on average across all of it. Taylor knows everything about the origin and may know nothing at $x = 2$. Fourier knows the general shape everywhere and is worst precisely at the sharp features, where its overshoot never goes away.

Which is better depends entirely on the question, and the pairing is the point: a function has a local description and a global one, they contain different information, and neither is more fundamental. The habit worth carrying is to ask which of the two a problem is really posing — because the answer determines which of them will be tractable, and choosing the wrong one is how a routine calculation becomes impossible.