1) $\displaystyle\frac12, \, \frac34,\, \frac78,\,
\frac{15}{16}, \, \frac{31}{32},\ldots$ converges to 1. 

2) The digits of $\pi$, namely 3, 1, 4, 1, 5, 9,
$\ldots$, diverge. 

3) The decimal approximations of $\pi$, namely 3, 3.1,
3.14, 3.141, $\ldots$, converge to $\pi$. 

4) Arithmetic
sequences, which are sequences where the difference between successive terms is
constant ($(a_{n+1}a_n)$ is constant), such
as 2, 5, 8, 11, 14, $\ldots$, always diverge (unless the
terms are all the same). 

5) The Fibonacci sequence 1, 1, 2, 3, 5, 8, $\ldots$
diverges. 

6) The sequence $\displaystyle 2, \, 1,\, \frac12,\,
\frac14,\,\frac18,\, \frac{1}{16}, \ldots$ is an example
of a geometric sequence (see below; here, $a=2$ and
$r=\tfrac 1 2$). This sequence converges to 0. 

7) Geometric
sequences are sequences where the ratio of successive terms is constant.
They look like$$\{ar^n\}_n= a,\,ar,\,ar^2,\,ar^3\ldots$$with
the constant ratio $r$. A geometric sequence converges
if $1<r\le1$ and diverges otherwise. (We can see
that if $r=1$, this sequence is the constant sequence,
$a,a,a,\ldots$, which converges to $a$.)

