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I came across this problem in a Quant finance prep test book. This looks a bit like a geometric series, a bit like an arithmetic series, and a bit like a recursive function.

$ f = \frac{1}{1 + \frac{1} {1 + ...}} $

Which I would rewrite as, $ f = \frac{1}{1 + f} $ , however, this might not be valid.

How do I solve this problem? And what is the name of this sort of problem?

jbuddy_13
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    see TheEpicCarrot7's answer. In other words, your analysis is fine, if you know that the continued fraction converges to a limit. So the evaluation reduces to proving that. If you have a strictly increasing or strictly decreasing bounded sequence, then the sequence must be convergent. So the evaluation reduces to proving that the sequence is both bounded and either strictly increasing or strictly decreasing. – user2661923 Apr 12 '21 at 09:08

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It is called a continued fraction, and this one, in particular, is equal to $\frac{\sqrt{5}-1}{2}$, which we get via the substitution you mentioned.

In order to show this is valid, we can simply show via induction that if we make a fraction of this form k fractions deep (e.g. if k=1 then the fraction would be 1/1, if k=2 then it would be 1/(1+1/1)), etc.), and this fraction is limited between two bounds, then the next fraction would also need to be in between those two bounds. In this case, we can show that a=1/(1+1/(1+...)) with depth k is between 0 and 2, then 1/(1+1/(1+...))=1/(1+a) with depth k+1 is between 1/(1+2)=1/3 and 1/(1+0)=1, and since 1/1 is between 0 and 2 the base case is good and we are done.

Andrew Carratu
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