Limits of sequences connected with real and complex exponential

Question

Let us denote $S_{n}(x)=1+\frac{x}{1 !}+\frac{x^{2}}{2!}+ ... + \frac{x^{n}}{n!}$.

1. How could be calculated the limit $$L(x)=\lim_{n\to \infty}\frac{S_{n}(n x)}{e^{n x}}=\lim_{n\to \infty}\frac{1+\frac{nx}{1 !}+\frac{(nx)^{2}}{2!}+ ... + \frac{(nx)^{n}}{n!}}{e^{n x}}, x\ge 0,\,\,\,\ ?$$

2. Similar question, when $x\ge 0$ above is replaced by $z\in \mathbb{C}$ ?

3. More general, if $r_{n}$ is a sequence of real numbers with $r^{n}\searrow 1$, what are the limits $\lim_{n\to \infty}\frac{S_{n}(n r_{n}x)}{e^{n x}}$ for $x\ge 0$ and $\lim_{n\to \infty}\frac{S_{n}(n r_{n}z)}{e^{n z}}$ for $z\in \mathbb{C}$ ? (Are they equal with the limits from the above points 1) and 2) ?

Initially, my intuition told me that probably that the limit $L(x)$ is equal to one, for any $x\ge 0$. But thinking better, my opinion is that the limit $L(x)=0$, for all $x\ge 0$. In support to this guess, for example, for $x=1$ I have calculated $\frac{S_{n}(n)}{e^{n}}$ for several consecutive values of $n$ and it appeared to me that it forms a decreasing sequence. In the general case, I have tried to use the Stolz-Cesaro lemma to the ratios $\frac{S_{n}(nx)}{e^{n x}}$ and $\frac{e^{nx}}{S_{n}(nx)}$, but it did not work. Also, I have tried to estimate $|S_{n}(nx)-e^{nx}|$ by using the Lagrange form of the remainder for Taylor series, but again I was not able to get any conclusion.In the complex case, the situation seems to be more intricated. Indeed, for $z=i$, we get $$\frac{S_{n}(n i)}{e^{n i}}=\frac{S^{(cos)}_{n}(n)+iS^{(sin)}_{n}(n)}{cos(n)+isin(n)},$$ where $S^{(cos)}_{n}$ and $S^{(sin)}_{n}$ represents the partial sums of order $n$ from the series development of cosine and sine functions. The limit with $n\to \infty$ in this case looks more tricky, as do not exist the limits $\lim_{n\to \infty}cos(n)$ and $\lim_{n\to \infty}sin(n)$.

Answer 1

This limit equals $0$ when $z$ is inside certain closed curve $\gamma$ and equals $\infty$ when $z$ is outside $\gamma$. This curve $\gamma$ is the boundary of the region $\{z:|z|<1,|ze^{1-z}|<1\}$, and it is called the Szego curve.

This follows from a result of Szego which gives a precise asymptotic behavior of the ratio $e^{-nz}S_n(nz)$.

G. Szego, Uber eine Eigenschaft der Exponentialreihe, Berlin Math. Ges. Sitzunsber. 23 (1924), 50-64.

If you have no access to this journal or do not read German, just type on Google: "Zeros of partial sum of the exponential" and you will see about 10 papers on this, most of them freely available in English.