在 2003/12/23 02:11:27 於未來最舊小棧留下的習題，終於解了。
題目 Link to heading
Define the sequence if numbers $a_n$ by
$$ a_0 = 1, a_1 = 1 + \frac{1}{1+a_0}, \cdots, a_n = 1 + \frac{1}{1+a_{n-1}} $$
Show that $a_n$ is convergent sequence. Find the limit.
證明 Link to heading
Let $A_0 = a_0$ and $A_n= a_n - a_{n-1}$, then
$$ \frac{A_n}{A_{n-1}} = \frac{-1}{(1+a_{n-1})(1+a_{n-2})} $$
It’s easy to see that $a_i\geq 1$, then $|A_n|\leq \frac{|A_{n-1}|}{4}$.
So $A_n$ is an alternating series with decreases monotonically and limit is 0.
By alternating series test, we have $a_n = \sum\limits_{i=0}^{n} A_n$ converges.
It’s easy to check $\lim\limits_{n\to\infty}a_n = \sqrt{2}.$
後記 Link to heading
當年沒有看出 $|A_n|\leq \frac{|A_{n-1}|}{4}$ 所以證不出 limit is 0。