Is there a known example of an infinite sum of finitely nested radicals that evaluates to a given value?
Or an infinite sum of convergents of an infinite continued fraction? The finitely nested radical bears the same relationship to the infinitely nested radical that the convergent bears to the infinite continued fraction, in case there is ambiguity.
If there is, is it possible that someone would kindly provide a reference?
Edit: With convergence to be assured, where necessary, by any suitable factor like factorials or powers etc., to diminish the convergents rapidly enough. . To rephrase the question, are there any examples of such sums involving nested radicals or convergents of continued fractions?