Examples of Cauchy sequences
In $\mathbb{R}$, it is true that every Cauchy sequence is convergent and vice-versa. After introducing the Cauchy sequence, usually, the explicit examples stated in almost all the books (and notes) are
constant sequence, $(\frac{1}{n})$, and any convergent sequence
The sequence $x_1=1$ and $x_{n+1}=1+\frac{1}{x_n}$.
Here the limits of the sequences are easy to compute.
Q.0 Are there any other typical examples of Cauchy sequences, which, from their expression, do not look convergent (or Cauchy)?
Q.1 Are there examples of Cauchy sequences, whose limits are not easy to find, or we can only say that it is Cauchy, without telling its limit?
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$\begingroup$Regarding Q1, you can easily encode open problems into the precise value of a sum which always converges, no matter the resolution of the open problem. E.g. let $A(n)=1$ if $n$ is a Fermat prime, and $A(n)=0$ otherwise. Then $L=\sum_{n=0}^\infty \frac{A(n)}{2^n}$ converges. By replacing $A(n)$ with $1$s, we see $L$ is trivially at most $2$. Actually it's much smaller...but how much smaller? If you could compute $L$ exactly, then writing $L$ in binary essentially gives you a list of all Fermat primes (of which we only know 5 exist, but it could in principle be an infinite list).
$\endgroup$ $\begingroup$Q0. There is a whole class of metric spaces in which Cauchy sequences may not converge to some element inside the metric space. We call those metric spaces "Incomplete metric spaces".
As an example, consider the metric space $(\mathbb{Q} , \rho)$ where $\mathbb{Q}$ is the set of rationals, and the metric $d := |x - y|~$ for all $x , y \in \mathbb{Q}$ . It can be easily verified that $d$ is indeed a metric.
Now, consider the sequence $\{a_n\}$ in $\mathbb{Q}$ defined by $a_0 = 1$ , and $a_{n+1} = 2 \cdot \dfrac{a_n + 1}{a_n + 2}$ .
Observe that $\{a_n\}$ is strictly increasing, and $a_n \leqslant 2~$ for all $~n \in \mathbb{N}$ . So, it converges to some point, say $L$ . But, taking $n \to \infty$ , we get that $\lim_{n\to \infty} a_n = \sqrt{2}~ \not\in \mathbb{Q}$ . Now, $\{a_n\}$ is a Cauchy Sequence in $\mathbb{Q}$ which doesn't converge in $\mathbb{Q}$ .
Q1. As @Bumblebee has mentioned in the comment section, take $\{a_n\}$ in $\mathbb{R}$ such that for all $n \in \mathbb{N}$ , $$a_n = 1 + \dfrac12 + \dfrac13 + \cdots + \dfrac1n - \log n$$ This is a Cauchy sequence which converges to the limit $\gamma$ , popularly known as the Euler - Mascheroni Constant. One can easily prove that this sequence converges, i.e. $\lim_{n \to \infty} a_n$ exists and is finite. But it's very difficult to show that the limit is precisely $\gamma$ .
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