1. ## Unbounded Sequence

Define a sequence s(sub n) by s(sub 1) = 1, s(sub n+1) = (s(sub n) )+ (s(sub n)/(1+s(sub n)^2)) for n greater than/equal to one. Show this sequence is unbounded.

2. Originally Posted by kjwill1776
Define a sequence s(sub n) by s(sub 1) = 1, s(sub n+1) = (s(sub n) )+ (s(sub n)/(1+s(sub n)^2)) for n greater than/equal to one. Show this sequence is unbounded.
It's an increasing sequence. So if it is bounded then it will converge. Show that this can't happen.

3. Originally Posted by Opalg
It's an increasing sequence. So if it is bounded then it will converge. Show that this can't happen.
How would I go about showing that this sequence can't converge?

4. Originally Posted by kjwill1776
How would I go about showing that this sequence can't converge?
Suppose that it does converge, to a limit $l$. Let $n\to\infty$ in the recurrence relation $s_{n+1} = s_n + s_n/(1+s_n^2)$, and you see that $l = l + l/(1+l^2)$. Therefore $l=0$. But that is impossible, because $s_1=1$, and the sequence is increasing, so its limit cannot be less than 1.