Evaluating Ramanujan’s Nested Radicals: A Sequential Approach

Rewriting a nested radical in this form is commonly known as denesting. Denesting a nested radical is often very difficult. Many papers have been written on simplifying radicals. For example, consider the papers by Landau [3], Osler [4], and Zippel [6], and the references contained therein. The great Indian mathematician Srinivasa Ramanujan Aiyangar also contributed a lot in this direction. In his unique way, Ramanujan observed several striking relationships among certain nested radicals. Here are some examples (note that the final example uses the standard notation for a continued fraction):