6
$\begingroup$

QUESTION:
Show that the sequence {$\sqrt{5},\sqrt{5+\sqrt{5}},\sqrt{5+\sqrt{5+\sqrt{5}}},\sqrt{5+\sqrt{5+\sqrt{5+\sqrt{5}}}},....$} is convergent and it converges to $\left(\frac{1+\sqrt{21}}{2}\right)$.

MY ATTEMPT:
The sequence takes the form of the recurrence $x_n=\sqrt{x_{n-1}+5}$. But neither can I show it to be monotonic increasing nor bounded. Once I have shown it to be convergent, I know how to find and show the limit. I have successfully done it too. But I cannot prove the convergence.

ONLY HINTS required.

P.S. Do not use Cauchy principle or any complicated test. I want the answer to be based on monotonicity and boundedness. For the problem belongs to that chapter only.

$\endgroup$
  • $\begingroup$ $\sqrt{a}<\sqrt{b}$ if and only if $a< b$ $\endgroup$ – Chinny84 Nov 10 '15 at 7:34
  • $\begingroup$ @Chinny84 How do I use this to prove monotonicity for large values of n? Induction perhaps? $\endgroup$ – SchrodingersCat Nov 10 '15 at 7:36
  • 2
    $\begingroup$ Hint: If $x_0<x_1$, can you compare $x_1$ and $x_2$? Which one is greater? $\endgroup$ – Did Nov 10 '15 at 7:38
  • $\begingroup$ @Did Your hint points to induction, I believe? $\endgroup$ – SchrodingersCat Nov 10 '15 at 7:44
  • $\begingroup$ INDUCTION and again INDUCTION $\endgroup$ – Bumblebee Nov 10 '15 at 8:46
6
$\begingroup$

Hints:

  • use induction
  • if $a_n \gt a_{n-1}$ what can you say about $\sqrt{5+a_n}$ compared with $\sqrt{5+a_{n-1}}$?
  • if $a_n \lt 4$ for example, what can you say about $\sqrt{5+a_n}$?
$\endgroup$
  • $\begingroup$ First bullet is for monotonicity, right? Now for 2nd bullet, when I know that $a_n>a_{n-1}$ why do I need to prove the root part? And for 3rd bullet, how do I say that $a_n<4$....intuition? $\endgroup$ – SchrodingersCat Nov 10 '15 at 7:54
  • 3
    $\begingroup$ You might want to show $a_{n+1} =\sqrt{5+a_n} \gt a_n$ as part of a proof by induction of monotonicity. Similarly you might want to show $a_{n+1} =\sqrt{5+a_n} \lt 4$ as part of a proof by induction of boundedness. As initial steps, you will need to show $\sqrt{5+\sqrt{5}} \gt \sqrt{5}$ and similarly $\sqrt{5} \lt 4$. $\endgroup$ – Henry Nov 10 '15 at 7:58
1
$\begingroup$

Have you thought about using functional approach? consider $f(x) = \sqrt{x+5}$, and show $f$ is increasing. This takes care of the monotonicity of $a_n$'s, and you can show $a_n < 3$ by induction.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.