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I'm working on some calculus exercises and came across this one. It is familiar yet not the same as the $\mathcal e$ one. The question states:

Given $X_0 = 1$ and $X_{n+1} = 1+\frac{1}{X_n}$, for $n \geq 0$.

Show that it converges and find its limit.

So I tried to approach this by looking at its behave. Does $\frac{1}{X_n}$ converge to $0$ ? But it's not trivial answer. I also tried to look at $X_{n+1} $ expressed as a function of $X_0$ but was hard to see if I could simplify it in one-line formula. Any ideas?

metamorphy
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Zuil Happy
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1 Answers1

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Let $\phi=\dfrac{1+\sqrt{5}}{2}$ and $Y_n=X_n-\phi$; then $$|Y_{n+1}|=\left|1+\frac{1}{Y_n+\phi}-\phi\right|=\left|\frac{(1-\phi)Y_n}{Y_n+\phi}\right|\leqslant|(1-\phi)Y_n|,$$ because $Y_n+\phi=X_n\geqslant 1$ clearly. As $|1-\phi|<1$, we have $\displaystyle\lim_{n\to\infty}Y_n=0$.

metamorphy
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