Maclaurin series

What you are looking at

The Maclaurin polynomial of degree $n$ is the polynomial that agrees with $f$ and its first $n$ derivatives at $x = 0$: $P_n(x) = f(0) + x f'(0) + \dfrac{x^2}{2!}f''(0) + \cdots + \dfrac{x^n}{n!}f^{(n)}(0)$. In the visualiser the curve is blue and $P_n$ is yellow. Raising $n$ matches one more derivative, and the yellow curve stays close to the blue one over a wider stretch around the origin.

How close the two are at a chosen point is shown by the vertical gap at $x_0$, drawn as a yellow segment and given in the panel as $|f(x_0) - P_n(x_0)|$. For $e^x$, $\sin x$ and $\cos x$ the series converges for every real $x$, although any fixed $P_n$ still peels away far from the origin. For $\ln(1+x)$ and $\arctan x$ the series converges only on a bounded interval, shaded green; outside it, adding terms does not close the gap.

Try this

1. 1With $f(x) = e^x$ and $x_0 = 1$, step $n$ up from $1$: $P_n(1)$ runs $2,\ 2.5,\ 2.667,\ 2.708, \ldots$ and closes in on $e \approx 2.718$.
2. 2Choose $\sin x$ and press Play. The polynomial only changes on odd values of $n$, because the Maclaurin series of $\sin x$ has no even powers.
3. 3Choose $\ln(1+x)$, drag $x_0$ to $2$ and raise $n$. A banner reports that the point is outside the interval of convergence, $-1 < x \le 1$, and the error grows as $n$ rises.
4. 4Choose $\arctan x$ and set $x_0 = 1$, the edge of the interval. Even at $n = 15$ the error is still about $0.03$: the series converges there, but slowly.

Mistakes students make

• Leaving out the factorials. The coefficient of $x^n$ is $\dfrac{f^{(n)}(0)}{n!}$, not $f^{(n)}(0)$.
• Assuming more terms always help. Outside the interval of convergence, as for $\ln(1+x)$ with $x > 1$, the partial sums do not approach $f(x)$ at all.
• Confusing the degree with the number of terms. For $\sin x$, $P_3(x) = x - \dfrac{x^3}{6}$ has only two non-zero terms, and $P_4$ is the same polynomial.

Where it is taught

• IB Maths AA HL 5.19 · Maclaurin series
• AP Calculus BC Unit 10 · Taylor polynomial approximations, interval of convergence and Maclaurin series (10.11, 10.13, 10.14)