Acta Math. Acad. Paed. Nyíregyháziensis
15 (1999), 35-40
www.bgytf.hu/~amapn

On the limit of a sequence

Z. László and Z. Vörös

Abstract:

The object of this article is to examine the sequence

$$a_n={\displaystyle{\sum_{i=0}^{n} {n^i \over {i!}}} \over e^n}$$

well known from probability theory. We prove that the sequence is bounded, strictly monotonously decreasing, and $\mathop {\hbox{lim}}_{n \to \infty}a_n={1 \over 2} \ .$The last two statements are proved by analytical means. Finally, a modification and a generalization of (an) will be mentioned, and the sketch of a second analytical proof for the original limit will be given.

 

Mathematics Subject Classification. 26A12, 26A42, 40A05.

Key words and phrases. Limit, probability, analytical means.