Taylor Series: Interactive Visualization

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Taylor series in this visualization

The Taylor series (Taylor expansion) of a function $f$ at a point $a$ is the power series built from its derivatives there; at $a = 0$ it is also called the Maclaurin series. The center is also written $x_0$ . Its partial sum of degree $n$ is the Taylor polynomial $P_n$ , the only polynomial of degree at most $n$ whose value and first $n$ derivatives at $a$ equal those of $f$ :

Taylor polynomial

P_n(x) = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!}\,(x-a)^k.

The ink curve is $f$ , the yellow curve is $P_n$ , and the blue point marks the center $a$ . The yellow bar on the $x$ -axis is a measurement: the widest interval $|x - a| < r$ on which $|f(x) - P_n(x)| < 0.01$ , found by checking points 0.0025 apart on both sides of $a$ and refining the first miss, and shown rounded down. The blue brackets sit at $a \pm R$ , where R is the radius of convergence: the series converges for $|x - a| < R$ and diverges for $|x - a| > R$ . For the seven functions here, the sum inside that interval is $f$ itself, and $R$ is the distance from $a$ to the nearest point where $f$ breaks down, a complex point included.

How far $P_n$ is from $f$ at a given $x$ is measured by the remainder. If $f$ has $n + 1$ derivatives on the whole interval between $a$ and $x$ (so that interval must not cross a singularity, such as the pole of $1/(1 - x)$ at 1), then

Remainder (Lagrange form)

f(x) - P_n(x) = \frac{f^{(n+1)}(\xi)}{(n+1)!}\,(x-a)^{n+1}\quad\text{for some } \xi \text{ between } a \text{ and } x.

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Things to notice

Inside the radius the partial sums converge to $f$ here, so at each such $x$ the error tends to 0 as $n$ grows, though not necessarily at every single step. Outside it the terms do not tend to 0 and $P_n(x)$ does not settle. The bar only approaches $R$ : it is measured on the screen, while $R$ comes from the formula.
A function can be smooth on the whole real line and still have a finite radius. $1/(1 + x^2)$ is undefined at the complex points $\pm i$ , so its series about $a$ converges only for $|x - a| < \sqrt{1 + a^2}$ .
At the ends of the interval anything can happen: the series of $\ln(1 + x)$ at 0 converges at $x = 1$ (to $\ln 2$ ) but not at $x = -1$ ; that of $1/(1 - x)$ converges at neither end, that of $\sqrt{1 + x}$ at both.
At $a = 0$ , $\sin x$ has only odd powers and $\cos x$ only even ones, so $P_{2m}$ and $P_{2m-1}$ of the sine are the same polynomial. n → ∞ skips such zero terms, so every step changes the curve.
Convergence alone does not make the sum equal $f$ . The function equal to $e^{-1/x^2}$ for $x \ne 0$ and 0 at 0 has every derivative 0 at 0, so its Taylor series there is 0 for all $x$ .

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Where it is used

Physics keeps the first terms near a point: $\sin\theta \approx \theta$ for a pendulum’s small swings, and $(1 - v^2/c^2)^{-1/2} \approx 1 + v^2/(2c^2)$ turns relativistic kinetic energy into $\tfrac12 mv^2$ at low speeds.
The remainder tells how many terms a computation needs: summing $1/k!$ from $k = 0$ to $10$ , which is $P_{10}(1)$ for $e^x$ , already gives $e$ to within $3 \times 10^{-8}$ .

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Controls

Turn the Degree knob, or press n → ∞ to add the next non-zero term, one at a time, up to $n = 30$ (pressing again at the top replays from 0). Drag the blue point or the marker on the axis to move the center; the key centers jump to exact values.
The strip below the graph shows $|f - P_n|$ on a logarithmic scale across the graph. Presentation mode opens a full-screen view for projectors (Space adds terms, ← → change the degree, H hides the radius).

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Taylor series in this visualization

Things to notice

Where it is used

Controls

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