# Weird Product Series

Evaluate the series:

$$\sqrt{1+z} \cdot \sqrt{1+z\sqrt{1+z}} \cdot \sqrt{1+z\sqrt{1+z\sqrt{1+z}}}\cdots\text{upto N terms}$$

How to find the general formula for $N$ terms?

• Is $z$ a complex number? – Alex Silva Jun 19 '18 at 12:49
• @AlexSilva No a variable – David Jun 19 '18 at 12:50
• @David Variables are numbers. The question remains: what possible numbers could $z$ be? Is it restricted to non-negative real numbers, for instance? – Arthur Jun 19 '18 at 12:52
• Z belongs to Real Number Set – David Jun 19 '18 at 12:53
• For the product to converge you need $a_{n+1}=\sqrt{1+za_n}$ to have limit $1$. So $1=\sqrt{1+z}$, which means $z=0$ is the only option. – Arnaud Mortier Jun 19 '18 at 12:55