Plus And Minus

Relevant wiki: Nested Functions

$f(k) : = \dfrac k{1 - \dfrac k {1 - \dfrac k{1 - \dfrac k {\ddots }}}}$

We want to determine whether $f(1)$ converge or not.

Suppose $f(k) \to L$ for some finite real $L$ , then $L = \dfrac k{1 - L }$ . Apply the quadratic formula gives $L = \dfrac{1 + \sqrt{1-4k}}2$ . So $f(k)$ only converges for $k\leq \dfrac14$ .

But for this question, $k = 1> \dfrac14$ . So $f(1)$ diverge.

---

This question was a direct reference from this discussion .

Square root of 3 + Square root of 3 + Square root of 3 ...... Like that how we solve same logic is applied here Assume that the denominator is x

1 + 1 ÷ 1 + 1÷ .....= x So equation is 1 + 1/x = x If we solve it we get X is 1+- √5/2.

doing the same for the other equation 1 - 1/x = x X^2 - x + 1 = 0 I got x is 1+- √-3/2 root of -3 is imaginary. Hence the answer is 67890. Real value does not exist.

Just list a few terms of the continued product. Some term will take the form $\dfrac 1 0$ .