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Math Help - Unbounded Sequence

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    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.
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    Quote Originally Posted by kjwill1776 View Post
    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.
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    Quote Originally Posted by Opalg View Post
    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?
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    Quote Originally Posted by kjwill1776 View Post
    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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