Generalization of Ramanujan Famous Nested Radicals to the nth Root and their Evaluation

Abstract

Srinivasa Ramanujan posed a problem on infinite nested radical of the square root in the Journal of Indian Mathematical Society in 1911. He had generated the problem years before in the form of an example illustrating a more general theorem. Of course, how we would figure the solution out without Ramanujan theorem was scarcely obvious. Generation of infinite nested radicals, the roots being of higher order i.e. proceeding from the cube roots to the nth root as well as their evaluation may seem to be somewhat complex. For this purpose a general solution in the form of a theorem needs to be provided whereby symmetric patterns of infinite nested radicals of any order of roots may be evaluated including that of Ramanujan involving square root. In our work, we have rendered the required solution.