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Math Help - Sequence

  1. #1
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    Sequence

    HI everybody,

    (\forall n \in \mathbb{N}) U_{n+1}=1+\frac{1}{1+U_n}  ,  U_0=1

    I showed that 1\le U_n\le \frac{3}{2}, but i don't know how to show that
    |U_{n+1}-U_n|\le \frac{1}{4}|U_n - U_{n-1}|

    Can you help me please???

    Thanks.
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  2. #2
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    Quote Originally Posted by lehder View Post
    HI everybody,

    (\forall n \in \mathbb{N}) U_{n+1}=1+\frac{1}{1+U_n} , U_0=1

    I showed that 1\le U_n\le \frac{3}{2}, but i don't know how to show that
    |U_{n+1}-U_n|\le \frac{1}{4}|U_n - U_{n-1}|

    Can you help me please???

    Thanks.


    Clearly U_n>0\,\,\,\forall n\in\mathbb{N} (you may want to prove this by induction), and then (U_{n-1}+1)(U_n+1)>4\,\,\,\forall\,n , so:

    \left|U_{n+1}-U_n\right|=\left|\frac{U_n+2}{U_n+1}-\frac{U_{n-1}+2}{U_{n-1}+1}\right|=\left|\frac{U_{n-1}-U_n}{(U_{n-1}+1)(U_n+1)}\right| ...

    Tonio
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  3. #3
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    YES, thank you very much, but why (U_{n-1}+1)(U_n+1)>4\,\,\???
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  4. #4
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    Quote Originally Posted by lehder View Post
    YES, thank you very much, but why (U_{n-1}+1)(U_n+1)>4\,\,\???


    Because in fact, for n\geq 2\,,\,\,U_n=1+\frac{1}{U_{n-1}+1}>1 ...

    Tonio
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