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Math Help - convergent sequence

  1. #1
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    convergent sequence

    Suppose x_1=\sqrt{2} and x_{n+1}=\sqrt{2x_n} for n\geq 1. Show that the sequence converges.
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  2. #2
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    Quote Originally Posted by Kat-M View Post
    Suppose x_1=\sqrt{2} and x_{n+1}=\sqrt{2x_n} for n\geq 1. Show that the sequence converges.
    x_1  < x_2  = \sqrt {2x_1 }  = \sqrt {2\sqrt 2 }  < 2

    \begin{gathered}<br />
  x_{N - 1}  < x_N  < 2 \hfill \\<br />
  2x_{N - 1}  < 2x_N  < 4 \hfill \\<br />
  \sqrt {2x_{N - 1} }  < \sqrt {2x_N }  < \sqrt 4  \hfill \\<br />
  x_N  < x_{N + 1}  < 2 \hfill \\ <br />
\end{gathered}

    So it increasing bounded above. Thus \left( {x_N } \right) \to L\;\& \,L = \sqrt {2L}
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