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.
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.
How would I go about showing that this sequence can't converge?
Suppose that it does converge, to a limit . Let in the recurrence relation , and you see that . Therefore . But that is impossible, because , and the sequence is increasing, so its limit cannot be less than 1.