limit of series

mascc · 2009-05-15 04:35:26

Does anyone know how to formally derive the limit to the following series for large n?

A_(n+1) = (1-epsilon)*(1 + A_n)

I can deduce using matlab that it is 1/epsilon - 1, but I would like to know how
to show it formally. 0 < epsilon < 1

Thanks

mascc · 2009-05-15 05:49:34

Never mind. It is easy if I set A_(n+1)=A_n and solve for A_n.

mathsyperson · 2009-05-15 07:45:55

That's not necessarily enough. Doing that only shows that if the sequence is convergent, then that's what the limit is.

You can actually express the whole sequence (A_n) explicitly, rather than in terms of recurrence:

Then since 0<ε<1, all appearances of (1-ε)^n will vanish as n tends to infinity and you're left with the required limit.

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