<s>[INST] Is the sequence $\left\{\frac{2^n}{n!}\right\}$ convergent? If so, what is the limit? Is the sequence $\left\{\frac{2^n}{n!}\right\}$ convergent? If so, what is the limit? 
$$ \frac{2^n}{n!} - 0 = \frac{2^n}{n!} < \frac {2^n}{n} <\; ? < \epsilon$$
I dont know  how to simplify $\frac{2^n }{ n}$.
I cannot just do $\frac{2^n}{n}<\epsilon$ right ? 
 [/INST] For $\; n \geq 3 \;$ we have
$$\frac{2^n}{n!} \;\; = \;\; \frac{2}{1} \cdot \frac{2}{2} \cdot \frac{2}{3} \cdot \frac{2}{4} \cdot \frac{2}{5} \cdot \frac{2}{6} \cdot \;\; \cdots \;\; \cdot  \frac{2}{n} \;\; \leq \;\; \frac{2}{1} \cdot \frac{2}{2} \cdot \left(\frac{2}{3}\right)^{n-2} \;\; \longrightarrow \;\; 0$$
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