I ended up with a task at the end of my post last month: Resting with ‘Nested Radicals’

I have figured out that is a diverging series. But now I am struck with another problem:

What is the corresponding sequence for nested radical series?

In general, I am given a sequence and I write the corresponding series and check its convergence/divergence. But what if I am given a series without a general term (though wrong in many cases, as a given pattern of finite terms may be written as many different general terms.). For nested radicals I believe that there is only one general term and without this general term I will not be able to write terms of corresponding sequence (as in case of nested radicals I simply get partial sums). So I have a bigger question to ask:

Does there exists a sequence corresponding to any given series?