5.7. Monotone and Bounded SequencesIn the last chapter the limit theorems provided a comfortable and quick method to decide on the convergence of a sequence. They are however applicable to only a small fraction of sequences as they have to meet a certain structure. So it is quite sensible to look for further covergence criterions. We take up again monotony and boundedness. Considered separately both properties are not or only sparsely related to convergence. Combining them surprisingly leads us to a new and useful criterion. Compared to the limit theorems however there is a slight disadvantage as we won't get any information on the value of the limit. Luckily there are some tricks to overcome this insufficiency. Furthermore the new criterion is only valid in the reals and not applicable with sequences e.g. in $\mathbb{Q}$ (cf. [5.7.11]).
Using the new criterion 'monotone and bounded' is always a two step task. First we solely prove the convergence and second calculate the limit. Our first example classifies all the sequences of the $({q}^{n})$ type.
Consider:
With the new criterion we can stock up the known convergences $\frac{1}{{n}^{k}}\to 0$.
The next example is a classical one. It introduces one of the most important mathematical constants, the number e.
Consider:
Using the boundedness of $({(1+\frac{1}{n})}^{n})$ e.g. above by 4 enables us to prove the convergence of another, nonelementary sequence.
The convergence of $(\sqrt[n]{n})$ entails in further results. For $1\le a\le n$ we have $1\le \sqrt[n]{a}\le \sqrt[n]{n}$ so the nesting theorem [5.5.8] provides
From that the same is also true for $0<a<1$: $\sqrt[n]{a}=\frac{1}{\sqrt[n]{\frac{1}{a}}}\to \frac{1}{1}=1$. The remainder examples now emphasize on the importance of the 'monotone and bounded' criterion for recursive sequences.
The example to follow is a valuable tool when calculating square roots approximatively. It is an ancient procedure as the name suggests.
Consider:
