On the zeros of the partial sums of $\cos(z)$ and $\sin(z)$

Abstract

Let \(s_{n}(z) = \sum_{k=0}^{n} \frac{z^{k}}{k!}\) denote the \(n\)-th partial sum of the exponential function. Carpenter et al. (1991) [1] studied the exact rate of convergence of the zeros of the normalized partial sums \(s_{n}(nz)\) to the so-called Szego-curve \(\) Here we apply parts of the results found by Carpenter et al. to the zeros of the normalized partial sums of \(\cos(z)\) and \(\sin(z)\;\).