4 ms·
I'm almost with you, but you need a bit more. To show what we want, we must prove the stronger condition that F(n)φ - F(n+1) → 0. Merely knowing that F(n+1)/F
by adrianratnapala 10y ago
I'm almost with you, but you need a bit more.
To show what we want, we must prove the stronger condition that F(n)φ - F(n+1) → 0. Merely knowing that F(n+1)/F(n) → φ only shows that the difference grows more slowly than F(n).
- winstonsmith 10y ago>To show what we want, we must prove the stronger condition that F(n)φ - F(n+1) → 0. Noting that {F(n+1)/F(n)} is the sequence of convergents of the continued fraction expansion of φ, your stronger condition follows from Theorem 5 at https://en.wikipedia.org/wiki/Continued_fraction https://en.wikipedia.org/wiki/Continued_fraction .
- deleted 10y ago[deleted]