Question

Does the sequence a1=1, a(n+1) = (an -1)/(an+1) converge?

Answer 1

As defined, the sequence is not convergent.

$\begin{case}a_1=1\\a_{n+1}=\dfrac{a_n-1}{a_n+1}&\text{for }n\ge2\end{cases}$
$\implies a_2=\dfrac{1-1}{1+1}=0$
$\implies a_3=\dfrac{0-1}{0+1}=-1$
$\implies a_4=\dfrac{-1-1}{-1+1}=-\dfrac20$

but the last term is undefined.