Standard

This is further continuation of stream of thoughts from my blog post: Nested Radical Sequences?

Just another idea against  representation of nested radical sequences came to my mind:

I know that, $\sqrt{1+\sqrt{1+\sqrt{\ldots}}}=\phi$ where $\phi$ is our famous golden ratio. But there is a theorem concerning series:

If $\sum_{n=1}^\infty a_n$ converges then $\lim_{n \to \infty} a_n = 0$

But to be able to check validity of this theorem I should be able to find $a_n$ which seems to be non-existent for nested radical sequences.