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!!

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