Unbounded Sequence

• Apr 29th 2009, 02:54 PM
kjwill1776
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.
• Apr 30th 2009, 03:55 AM
Opalg
Quote:

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.
• May 3rd 2009, 06:48 PM
kjwill1776
Quote:

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?
• May 4th 2009, 01:30 AM
Opalg
Quote:

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.