377. It is known that the number $\boldsymbol{e}$ (the base of natural logarithms) is defined as the limit of the expression $\left(1+\frac{1}{n}\right)^{n}$ as $n$ increases without bound:

$$
e=\lim _{n \rightarrow \infty}\left(1+\frac{1}{n}\right)^{n}
$$

But one might reason as follows: if the exponent of a base slightly greater than 1 increases without bound, then the power also increases without bound, hence $e=\infty$ (see question 376). Or else: as $n$ approaches the limit, $\frac{1}{n}$ becomes zero, and therefore $1+\frac{1}{n}$ becomes 1; any power of 1 is 1; hence, $e=1$. Yet it is claimed that the transcendental number $e=2,71828 \ldots$ Where is the truth?
377. Of course, both conclusions drawn are unjustified, because the propositions about the limit of a power apply only to a power with a constant base (or $1+\alpha$, or 1), while the exponent increases, whereas here the base $1+\frac{1}{n}$ itself is variable. In more detailed algebra textbooks (or introductory differential calculus), it is proven: 1) that the expression $\left(1+\frac{1}{n}\right)^{n}$ constantly increases as $n$ increases; 3) that, despite increasing, it remains less than the number 3; 3) that, consequently, it tends to a certain limit less than three; 4) they calculate this limit to the desired degree of accuracy and obtain the specified value $e=2.71828 \ldots$