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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?

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  • $\begingroup$ Is $z$ a complex number? $\endgroup$ – Alex Silva Jun 19 '18 at 12:49
  • $\begingroup$ @AlexSilva No a variable $\endgroup$ – David Jun 19 '18 at 12:50
  • $\begingroup$ @David Variables are numbers. The question remains: what possible numbers could $z$ be? Is it restricted to non-negative real numbers, for instance? $\endgroup$ – Arthur Jun 19 '18 at 12:52
  • $\begingroup$ Z belongs to Real Number Set $\endgroup$ – David Jun 19 '18 at 12:53
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    $\begingroup$ 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. $\endgroup$ – Arnaud Mortier Jun 19 '18 at 12:55

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