Math, Mathematics, Numerical Analysis, Science.
the iteration actually tends to -√2; not a problem! Check: doing the numbers, we get N₉=-816 and D₉=577 that approximate -√2 (1.414213562373…) as -1.41421… with 6 digits’ precision; later N₁₃=-27720 and D₁₃=19601 producing -1.41421356… with 9 digits’ precision, and eventually N₁₇=-941664 and