Unbounded Sequence

**kjwill1776** · April 29th 2009, 01:54 PM

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.

**Opalg** · April 30th 2009, 02:55 AM

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.

**kjwill1776** · May 3rd 2009, 05:48 PM

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?

**Opalg** · May 4th 2009, 12:30 AM

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.

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