Find all x where that limit is true

Question

Let the following expression, with $n \in \mathbb{N}$

$$ T_n = \underbrace{\sqrt{x \sqrt{x \sqrt{x \dots \sqrt{x}}}}}_{\text{n times}} $$

It's easy to see that

$$\lim_{n \to \infty} T_n = x$$

Find all x where that limit is true

Answer 1

You may observe that $$ T_n = \underbrace{\sqrt{x \sqrt{x \sqrt{x \dots \sqrt{x}}}}}_{\text{n times}}=x^{\large \frac12+\frac1{2^2}+...+\frac{1}{2^n}}=x^{\large 1-\frac{1}{2^n}}.$$ Hence for any complex number with non negative real part $x$ you have $$\lim_{n \to \infty} T_n = x.$$