Taylor polynomial explorer

Pick a function, then set the degree $n$ and the center $a$. The curve in the text color is the function, and the colored curve is its Taylor polynomial $T_n$. Move the point $x$ with the slider or by dragging across the graph, and the readouts compare $T_n(x)$ with the true value $f(x)$. The shaded band is the interval where the full Taylor series converges. For $e^x$, $\sin x$ and $\cos x$ the series converges everywhere, so there is no band.

Things to try

• With $e^x$ centered at $0$, set $x = 1$ and raise the degree one step at a time. The polynomial value climbs toward $e = 2.7182818$, and by degree $10$ the error is down to about $2.73 \times 10^{-8}$.
• Switch to $\sin x$ centered at $0$ and go from degree $3$ to degree $4$. Nothing moves, because sine has no even powers in its series. Keep raising the degree and the polynomial follows the curve across a wider stretch each time. Even at degree $15$ it misses $\sin 6.5$ by about $0.16$.
• Pick $\ln(1 + x)$, set $x = 1.5$ and raise the degree. From degree $2$ on, every step makes the error larger, because $1.5$ lies outside the band. Back at $x = 0.5$ the error shrinks with every step.
• Stay with $\ln(1 + x)$ at $x = 1$, the right end of the band. The series still converges there, but slowly. At degree $15$ the error is still about $0.03$.
• Try $\frac{1}{1 + x^2}$ at $x = 1.5$. The function is smooth on the whole real line, yet the band stops at $1$ and the error grows with the degree. Now move the center to $a = 1$. The radius becomes $\sqrt{2} \approx 1.414$, the band now covers $1.5$, and the error shrinks instead.
• With $\sqrt{1 + x}$ centered at $0$, compare $x = 1$ with $x = 2$. At $x = 1$ the error shrinks slowly toward $0$. At $x = 2$ it grows with every step from degree $2$ on.
• Keep the degree fixed at $3$ and move the center of $e^x$ toward the point. At $x = 1$ the error is about $0.052$ with $a = 0$ and about $0.0048$ with $a = 0.5$.

What you are seeing

The Taylor polynomial of degree $n$ centered at $a$ matches $f$ and its first $n$ derivatives at $a$.

The explorer computes every coefficient from an exact formula for the derivatives. For example, every derivative of $e^x$ is $e^x$, and the derivatives of $\ln(1 + x)$ are

At $a = 0$ the readout writes the coefficients as exact fractions, and at other centers it rounds them to four decimals. Past degree $6$ it shows the first terms only.

The error readout is the remainder $R_n(x) = f(x) - T_n(x)$. Taylor’s theorem says it has the same shape as the next term, with the derivative taken at some $z$ between $a$ and $x$.

The factor $(x - a)^{n+1}$ is why the error shrinks so fast near the center and why moving the center closer to $x$ helps.

The band is the interval $(a - R, a + R)$, where $R$ is the radius of convergence of the full series. For $\ln(1 + x)$ and $\sqrt{1 + x}$ the series reaches only as far as the trouble at $x = -1$, so $R = 1 + a$. For $\frac{1}{1 + x^2}$ nothing goes wrong on the real line. The trouble sits at the complex numbers $\pm i$, where $1 + x^2 = 0$, and the radius is the distance from $a$ to them.

Inside the band the error goes to $0$ as the degree grows. Outside it the terms of the series grow, and the polynomials swing further from the function with every degree. The curve of $T_n$ is cut off at the edges of the window when it runs off the graph.

The series and the remainder bounds are on the calculus reference sheet, and the videos are on the Calculus course page.