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.

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- Apr 29th 2009, 01:54 PMkjwill1776Unbounded 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, 02:55 AMOpalg
- May 3rd 2009, 05:48 PMkjwill1776
- May 4th 2009, 12:30 AMOpalg
Suppose that it does converge, to a limit $\displaystyle l$. Let $\displaystyle n\to\infty$ in the recurrence relation $\displaystyle s_{n+1} = s_n + s_n/(1+s_n^2)$, and you see that $\displaystyle l = l + l/(1+l^2)$. Therefore $\displaystyle l=0$. But that is impossible, because $\displaystyle s_1=1$, and the sequence is increasing, so its limit cannot be less than 1.