The above challenging and interesting questions were posed by Srinivas Ramanujan (one who needs no introduction) in the Journal of Indian Mathematical Society, vol. 3, 1911.

You must be wondering what must be the answers to these questions.

Well the answer to the first question is 3 and for the second it is 4.

You must now be wondering how do you get these answers and how must have Ramanujan created such wonderful mathematical expressions?

Let us consider the below expression:

Will this continue forever? How can we be sure? Will this work for any number n?

Ramanujan had given the answers to the above questions in the Journal of Indian Mathematical Society, vol. 4, 1912.

I will derive and prove the above infinite nested radicals by a method different from Ramanujan's.

Consider the expression given below:

- (1)

Hence, we now have an important result:

From (1), we have

Now using result (2), we have

Putting this in (1) above, we get

Thus, it is proved.

Now, for n = 1, we have

And, for n = 2, we have

For the second question let us now consider the below expression:

 

Will this continue forever? How can we be sure? Will this work for any number n?

Consider the mathematical expression given below:

- (3)

Hence, we now have an important result:

From (3), we have

Now using result (4), we have

Putting this in (4) above, we get

Thus, it is proved.

Now, for n = 1, we have

And, for n = 2, we have

 

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