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Thread: Sequence Convergence

  1. #1
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    Sequence Convergence

    so the sequence is

    {$\displaystyle \sqrt2, \sqrt(2+\sqrt2),\sqrt(2+\sqrt(2+\sqrt2), ...$ }

    i need to prove by induction that the sequence is monotonic and bounded.
    i can see how its monotonic, but how do i prove its bounded? and what is induction?
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  2. #2
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    Quote Originally Posted by ixion124 View Post
    so the sequence is

    {$\displaystyle \sqrt2, \sqrt(2+\sqrt2),\sqrt(2+\sqrt(2+\sqrt2), ...$ }

    i need to prove by induction that the sequence is monotonic and bounded.
    i can see how its monotonic, but how do i prove its bounded? and what is induction?
    Work with $\displaystyle s_1=\sqrt2$, $\displaystyle n\ge2,~s_n=\sqrt{2+s_{n-1}}$.
    Use induction.
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  3. #3
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    thanks!
    but how would i prove it with induction?
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  4. #4
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    Show that $\displaystyle s_1\le s_2\le 2$.

    Show that if $\displaystyle s_N\le s_{N+1}\le 2$ then $\displaystyle s_{N+1}\le s_{N+2}\le 2$.
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