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- Thread starter FallenApple
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I'm writing a computer program to show convergence. Say the function is f(n)=n/(n+1). Now we know that the limit is 1. But its kinda cheating to use that knowledge in the program since that somewhat defeats to purpose of finding the limit numerically, so I just run the iterations up until the difference between consecutive terms is less than some threshold.

Then I return the iteration number and the current term of the sequence to see how fast it converges and get an approximation of the limit.

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They will eventually decrease but, without additional info, we cannot conclude anything from current differences about bounds on future differences. A function may appear to be asymptotically approaching a limit from above but then bump up significantly before finally approaching a limit.

eg consider

$$f(x)=e^{-x} + e^{-\frac{(x-1000)^2}2}$$

which has a Gaussian bump centred at ##x=1000## in what is otherwise an asymptotic, concave-up, slide towards a limit of 0 as ##x\to\infty##.

Even more pathological functions could be created that bump up infinitely many times. One possibility may be:

$$g(x)=

e^{-x} + \sum_{k=1}^\infty e^{-\frac{(x-k\cdot 1000)^2}2}$$

I have not proven that the function is well-defined, although I suspect it is.

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mathwonk

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