# Convergence of a sequence

**Question**

Limit of the sequence !!sqrt(2)!!, !!sqrt(2(sqrt2))!!, ... !!sqrt(2(sqrt2(sqrt2...)))!! is?

- 2
- !!sqrt2!!
- !!2sqrt2!!
- None

**Solution**

This problem can be solved by 10th and 11th class students using different approaches.

First let us assume the limit of the sequence be !!x!!

Then the next term of the sequence is !!sqrt(2x)!!

But since !!x!! is the limit of the sequence !!x=sqrt(2x)!!

So we get !!x^2=2x!!, !!x!=0!! so !!x=2!!

**Alternatively**

We can !!nth!! term of the sequence as !!2^(1/2+1/4+...+1/2^n)!!.

So the value of the this term as !!n->oo!! is !!2^(1/2/(1-1/2))=2!!