Does $\int _0^1 \cfrac{x}{x+\cfrac{x^{2}}{x^2+\cfrac{x^3}{x^3+\cdots}}} \, dx$ exist?

Question

$$\int_0^1 \cfrac x {x+\cfrac{x^2}{x^2+\cfrac{x^3}{x^3+\cdots}}} \, dx$$

Does this integral exist?

Any hint will be appreciated.

omg ur question said hint – Saketh Malyala May 01 '17 at 05:19 — Saketh Malyala, May 01 '17 at 05:19
does a solution count as a hint bc same – Saketh Malyala May 01 '17 at 05:19 — Saketh Malyala, May 01 '17 at 05:19

score 6 · Accepted Answer · answered May 01 '17 at 05:15

6

$\int _{0}^{1} \frac{x}{x＋\frac{x^{2}}{x^2＋\frac{x^{3}}{x^{3}＋…}}}dx$

Let's call that monstrous integrand $u$. If you notice, $u=\frac{x}{x+xu}=\frac{1}{1+u}$.

$u+u^2=1$, so $u^2+u-1=0$. This means $u=\frac{\sqrt{5}-1}{2}$.

$\int_0^1 \frac{\sqrt{5}-1}{2} \,dx = \boxed{\frac{\sqrt{5}-1}{2}}$

answered May 01 '17 at 05:15

Saketh Malyala

but when $x$ goes to $0$，does $\frac{x}{x＋xu}＝\frac{1}{1+u}$ still hold？ – z3wood May 01 '17 at 05:20
Short and sweet. – Andrew Tawfeek May 01 '17 at 05:20
i guess that's the point of a limit. @Goodboy – Saketh Malyala May 01 '17 at 05:21
ok，thanks a lot – z3wood May 01 '17 at 05:22
2

No this is wrong, in the recursion $x u = \frac {x^2}{x + \frac{x^2}{x^2 + ...}} \not= \frac {x^2}{x^2 + \frac{x^3}{x^3 + ...}}$ – Wei Zhong Dec 15 '20 at 09:42

1 Answers1