Category: Monotonic Sequence

Monotonic Sequence Definition

Sequence ${x}_{{n}}$ is called increasing if ${x}_{{1}}<{x}_{{2}}<\ldots<{x}_{{n}}<{x}_{{{n}+{1}}}<\ldots$, i.e. if ${n}'>{n}$ then ${x}_{{{n}'}}>{x}_{{n}}$.

For example, ${\left\{{1},{2},{3},{6},{7},{9},\ldots\right\}}$ is increasing sequence, while ${\left\{{3},{5},{8},{1},{5},{6},{7},\ldots\right\}}$ is not.

The Euler Number $e$

Now let's consider sequence ${x}_{{n}}={{\left({1}+\frac{{1}}{{n}}\right)}}^{{n}}$ and try to find its limit.

It is not very clear whether this sequence is monotonic or not.

So, to make sure that this sequence is increasing let's rewrite sequence using binom of Newton with ${a}={1}$ and ${b}=\frac{{1}}{{n}}$: