Question

does fibonacci converge ?

Answer 1

Nope. Do you recall the following:\[\bf \lim_{n \rightarrow \infty}\frac{ f _{n+1} }{ f _{n} }=\phi \approx 1.618...\] The ratio of the next and previous terms of the sequence is always greater than 1, but for a series to converge, the terms of the sequence must approach 0, but here that doesn't happen. Each term is larger than the one before and;\[\bf \lim_{n \rightarrow \infty} f_{n} \ne 0\]

Answer 2

@Jonask

Answer 3

yes the golden ratio
so it diverges since by ratio test this will mean
\[\huge \frac{ a_{n+1} }{ a_n } \ge 1\]

Answer 4

mhm