Complete Elementary Analysis of Nested radicals

This is a continuation of the series of summer projects sponsored by department of science and technology, government of India. In this project work, I have worked to collect and expand what Ramanujan did with Nested Radicals and summarized all important facts into the one article. In the article, there are formulas, formulas and only formulas — I think this is exactly what Ramanujan is known for.

This article not only deals with Ramanujan’s initial work on Nested Radicals but also develops a few new models and adds more information to it by catching the latest research in a very elementary way. The project was initiated first at in 2011.

Download the original PDF copy of the article (here) and let me know what you think about it.


Indian mathematical prodigy Srinivasa Ramanujan needs no introduction due to his remarkable contributions to various fields of mathematics.

In his brief lifetime, he discovered and published several theorems and formulas which even today surprise the leading math majors. This project is based on one of his very first contributions to mathematics, in form of some excellently patterned and formulized nested radicals.

This is a summary of the proper research about Ramanujan’s Nested Radicals, initiated first at and done rigorously in recent months.

We shall start with the basic definition of nested radicals and will end with the analytic properties. In between, we’ll explore how good Ramanujan’s
work on nested radicals go.

On an elementary basis, the theory of Ramanujan’s radicals is almost complete in reference and has hardly any chance of finding a research topic in it.

My aim is, however, to build strong basics in the topic at the instance and to do proper research in upcoming days by learning more about algebraic, analytic and
modular aspects of number theory.