Real zeros of sine Taylor polynomials

The more terms of Taylor series $\sin x = x-\frac{x^3}{3!}+ \frac{x^5}{5!}- \cdots$ we use, the more resemblance we see between the Taylor polynomial and the sine function itself. The first-degree polynomial matches one zero of the sine, and gets the slope right. The third-degree polynomial has three zeros in about the right places.

Third degree Taylor polynomial — Third degree, three zeros

The fifth-degree polynomial will of course have … wait a moment.

Fifth degree Taylor polynomial — Fifth degree, only one zero

Since all four critical points are in the window, there are no real zeros outside of our view. Adding the fifth-degree term not only fails to increase the number of zeros to five, it even drops it back to the level of ${T_1(x)=x}$ . How odd.

Since the sine Taylor series converges uniformly on bounded intervals, for every ${ A }$ there exists ${ n }$ such that ${\max_{[-A,A]} |\sin x-T_n(x)|<1 }$ . Then ${ T_n }$ will have the same sign as ${ \sin x }$ at the maxima and minima of the latter. Consequently, it will have about ${ 2A/\pi }$ zeros on the interval ${[-A,A] }$ . Indeed, the intermediate value theorem guarantees that many; and the fact that ${T_n'(x) \approx \cos x }$ on ${ [-A,A]}$ will not allow for extraneous zeros within this interval.

Using the Taylor remainder estimate and Stirling's approximation, we find ${A\approx (n!)^{1/n} \approx n/e }$ . Therefore, ${ T_n }$ will have about ${ 2n/(\pi e) }$ real zeros at about the right places. What happens when ${|x| }$ is too large for Taylor remainder estimate to be effective, we can't tell.

Let's just count the zeros, then. Sage online makes it very easy:

sineroots = [[2*n-1,len(sin(x).taylor(x,0,2*n-1).roots(ring=RR))] for n in range(1,51)]
scatter_plot(sineroots)

The up-and-down pattern in the number of zeros makes for a neat scatter plot. How close is this data to the predicted number ${ 2n/(\pi e) }$ ? Pretty close.

scatter_plot(sineroots,facecolor='#eeee66') + plot(2*n/(pi*e),(n,1,100))

The slope of the blue line is ${ 2/(\pi e) \approx 0.2342 }$ ; the (ir)rationality of this number is unknown. Thus, just under a quarter of the zeros of ${ T_n }$ are expected to be real when ${ n }$ is large.

The actual number of real zeros tends to exceed the prediction (by only a few) because some Taylor polynomials have real zeros in the region where they no longer follow the function. For example, ${ T_{11} }$ does this:

Richard S. Varga and Amos J. Carpenter wrote a series of papers titled Zeros of the partial sums of ${ \cos z }$ and ${\sin z }$ in which they classify real zeros into Hurwitz (which follow the corresponding trigonometric function) and spurious. They give the precise count of the Hurwitz zeros: ${1+2\lfloor n/(\pi e)\rfloor }$ for the sine and ${2\lfloor n/(\pi e)+1/2\rfloor }$ for the cosine. The total number of real roots does not appear to admit such an explicit formula. It is the sequence A012264 in the OEIS.

Real zeros of sine Taylor polynomials

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