A Practical Guide To Quantitative Finance Interviews 收录了一些量化面试中的经典题目. 本文介绍的是来自第二章的「Infinite Sequence」.

Infinite Sequence #

Problem #

If x ^ x ^ x ^ x ^ x … = 2 , what is x?

Solution #

记 $y = x^{x^{\dots}}$ , 则 $y = x^y$ , 故 $x^2=2$ , 因此 $x = \sqrt 2$ (排除 $-\sqrt 2$).

More about the sequence #

$x^{x^{\dots}} = y$ has an solution only when $y\in [\frac{1}{e},e]$ , which also means that $x^{x^{\dots}}$ is convergent only for $x\in [e^{-e}, e^{\frac{1}{e}}]$ .