Sunday, January 13, 2019

real analysis - Where does this sequence $sqrt{7}$,$sqrt{7+ sqrt{7}}$,$sqrt{7+sqrt{7+sqrt{7}}}$,.... converge?





The given sequence is $\sqrt{7}$,$\sqrt{7+ \sqrt{7}}$,$\sqrt{7+\sqrt{7+\sqrt{7}}}$,.....and so on.



the sequence is increasing so to converge must be bounded above.Now looks like they would not exceed 7. The given options are




  1. ${1+\sqrt{33}}\over{2}$


  2. ${1+\sqrt{32}}\over{2}$


  3. ${1+\sqrt{30}}\over{2}$


  4. ${1+\sqrt{29}}\over{2}$





How to proceed now.
Thanks for any help.


Answer



Trick: Let $X = \sqrt{ 7 + \sqrt{ 7 + ... } } $. We have $X = \sqrt{ 7 + X } $ and so $X^2 = 7 + X $. Now you solve the quadratic equation.


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