Nested Radicals & Limit

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.

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