A property of roots of the truncated series for $\sin(x)$

Let $p_n(x) = \sum\limits_{k=0}^n \frac{(-1)^kx^{2k+1}}{(2k+1)!}$

In other words, $p_n$ is the polynomial made of the first $n$ terms of the Taylor expansion of $\sin(x)$ around $x = 0$.

$\begin{align*} p_0(x) &= x\\ p_1(x) &= x - \frac{x^3}{6}\\ p_2(x) &= x - \frac{x^3}{6} + \frac{x^5}{120}\\ &\vdots\end{align*}$

I am interested in the roots of these polynomials. Each $p_n$ has some real roots and some complex roots. Given $n$, let $M$ be the maximum absolute value of all the real roots of $p_n$.

How can we show that every complex root $z$ of $p_n$ has $\vert \Re(z)\vert > M$?

Solution 1:

I refer you to paper 230 from Richard S Varga's collection of papers with the title 'Zeros of the partial sums of $\cos (z)$ and $\sin(z)$'.

An important step is to view the scaled (by degree) truncated taylor polynomial. If your polynomial $p(z)$ is of degree $n$ you would look at $p(nz)$. You can also google the 'Szegő Curve' which is given by $|ze^{1-z}|=1$ with $|z| \leq 1$. This is the curve that the scaled taylor polynomial of $e^z$ approaches as $n\rightarrow\infty$.

Otherwise the link to MO that Olivier posted is also very helpful.

And of course Olivier's suggestion to look at the original paper by Szegő, which contains a lot of this already as well (if you can get it and can understand german). Otherwise you should loook at varga's paper 221 - which is mostly about convergence of the taylor polynomials of $e^z$ to the Szegő Curve. So I referenced two papers, namely 230 and 221 - not the same one twice.

Solution 2:

Here's an interesting result that is related to your question.

Let $P_{n}(x)$ be the $n$th order Taylor polynomial for $\sin(x)$ about $x=0.$ Thus,

$$P_{1}(x) = P_{2}(x) = x,$$

$$P_{3}(x) = P_{4}(x) = x - {\tiny \frac{1}{6}}x^{3}, \;\; \mbox{etc.}$$

On each compact interval, these polynomials converge uniformly to $\sin(x)$ as $n \rightarrow \infty,$ so it follows that the number of zeros of $P_{n}(x)$ approaches $\infty$ as $n \rightarrow \infty.$

Let $Z(n)$ be the number of real zeros, counting multiplicity, of $P_{n}(x)$. Then

$$\lim_{n \rightarrow \infty} \frac{Z(n)}{n} \; = \; \frac{2}{\pi e}$$

A proof is given in the following 2 page paper by Rothe, which is on the internet (.pdf file). The proof given in this paper should be accessible to a fairly strong high school calculus student.

Frantz Rothe, Oscillations of the Taylor polynomials for the sin function, Nieuw Archief voor Wiskunde (5) 1 (2000), 397-398.

http://www.nieuwarchief.nl/serie5/pdf/naw5-2000-01-4-397.pdf

This result can also be found in the following paper (not mentioned by Rothe):

Norman Miller, The Taylor series approximation curves for the sine and cosine, American Mathematical Monthly 44 #2 (February 1937), 96-97.

Solution 3:

The roots of $\sin(z)$ can only be either symmetric real pairs, symmetric imaginary pairs or complex quartets. In the case of the polynomials from the Taylor expansion this reduces to real pairs and complex quartets.

Paper 230 proves the hypothesis that the modulus of the complex quartets is larger than the largest real root for the sine function as can be seen on the figure 2 for $60z$ and $61z$. This is what makes it impossible for these complex roots to converge as every next polynomial will have new and larger real roots.
For an example of a case where the complex quartets do converge take $\sinh(z^2)$