Calculus of Ramanujan’s Infinite Nested Radicals
Looking at an infamous problem posed by Srinivasa Ramanujan and an integral with a golden solution
One part of mathematics that I find fascinating however hard to grasp are infinite series, summations and nested radicals. Probably my first encounter with infinite nested radicals was from Srinivasa Ramanujan’s work. Arguably, his most infamous nested radical which he proposed to the ‘Journal of the Indian Mathematical Society’
Now to start let us walk through a solution provided by Ramanujan himself then we shall look at an integral containing a similar infinite nested radical.
Let f(n) be defined as
Then notice also that
Therefore we can rewrite our function as
You may have recognised that
Let’s obtain another equation for f(n+1) by letting n=>n+1 in (1)
From this we can substitute (2) into (1) and we get
Following this method for f(n+2), we obtain this equation
Repeating this method infinitely many times we get this final equation
Setting n=1 in (4) we get back to our original problem, therefore we can say
Next we will look to solve an integral that contains an infinite nested radical. So, here is the problem
When I first saw the integral, it looked quite daunting for someone who has limited calculus knowledge. Let’s first deal with the infinite radical on the denominator.
After a simple substitution we have obtained the value of the radical, which now we can use to simplify our problem. You may notice that I have only used the positive value of the square root when solving the quadratic, that is because there are no negative terms in the radical.
We can now rewrite our integral and after a u substitution we can get an a fascinating answer
I find it adds to the satisfaction of solving a problem when a mathematical constant turns up out of nowhere like φ and it is fascinating to see that these constants are behind so many parts of mathematics.
Thank you for reading
