 $\displaystyle\frac12, \, \frac34,\, \frac78,\, \frac{15}{16}, \, \frac{31}{32},\ldots$ converges to 1
 The digits of $\pi$, namely 3, 1, 4, 1, 5, 9, $\ldots$, diverge
 The decimal approximations of $\pi$, namely 3, 3.1, 3.14, 3.141, $\ldots$, converge to $\pi$.
Arithmetic sequences, namely sequences where $(a_{n+1}a_n)$ is constant, such as 2, 5, 8, 11, 14, $\ldots$, always diverge (unless they are constant).

 The Fibonacci sequence 1, 1, 2, 3, 5, 8, $\ldots$ diverges
 The sequence
$\displaystyle 2, \, 1,\, \frac12,\, \frac14,\,\frac18,\,
\frac{1}{16}, \ldots$ is an example of a geometric sequence. It converges to 0.
The geometric sequence $$\{cr^n\}_n= c,\,cr,\,cr^2,\,cr^3\ldots$$ converges if $\lvert r\rvert<1$ or if $r=1$, and diverges if $\lvert r\rvert>1$ or if $r=1$.

