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Thread: Increasing monotonic series

  1. #1
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    Increasing monotonic series

    Hey.

    I have recursive series:

    $\forall n \in \mathbb{N}, \ \ a_{n+1}=\sqrt{2+a_n}, \ \ a_1=t, \ \ -2\leq t \in\mathbb{R}$

    Prove:

    If $-2\leq t\leq 2$ then A(n) is defined to every natural n and the series A(n) is an increasing monotonic series and is bounded above.

    We have studied the material so far up to Cantor's lemma, but I don't know how to start here.

    I have begun by trying to prove that this series is at least an increasing-monotonic but I got that this inequality has to be true:
    $a_n^2<2+a_n$
    Which I don't know how to confirm.

    As to bounded above, have no idea how to think here on bounds...

    Thanks.
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  2. #2
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    Re: Increasing monotonic series

    Quote Originally Posted by CStudent View Post
    I have recursive series:
    $\forall n \in \mathbb{N}, \ \ a_{n+1}=\sqrt{2+a_n}, \ \ a_1=t, \ \ -2\leq t \in\mathbb{R}$

    Prove:
    If $-2\leq t\leq 2$ then A(n) is defined to every natural n and the series A(n) is an increasing monotonic series and is bounded above.
    There are mistakes in the statement. You write about a series and then proceed to define a sequence $a_n$. THEN you proceed further to ask about a function $A(n)$ without defining it.
    Is $\displaystyle A(n) = \sum\limits_{k = 1}^n {{a_k}} $ or is it simply $A(n)=a_n~?$ Is it a series or is it a sequence?
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  3. #3
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    Re: Increasing monotonic series

    first prove by induction that

    $\displaystyle a_n\leq 2$
    Thanks from CStudent
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  4. #4
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    Re: Increasing monotonic series

    Quote Originally Posted by Plato View Post
    There are mistakes in the statement. You write about a series and then proceed to define a sequence $a_n$. THEN you proceed further to ask about a function $A(n)$ without defining it.
    Is $\displaystyle A(n) = \sum\limits_{k = 1}^n {{a_k}} $ or is it simply $A(n)=a_n~?$ Is it a series or is it a sequence?
    It's a series
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  5. #5
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    Re: Increasing monotonic series

    Quote Originally Posted by Idea View Post
    first prove by induction that

    $\displaystyle a_n\leq 2$
    How do you realize at first glance that this series is bounded above by 2?

    Thanks.
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  6. #6
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    Re: Increasing monotonic series

    Quote Originally Posted by CStudent View Post
    How do you realize at first glance that this series is bounded above by 2?

    Thanks.
    we want to show that $\displaystyle a_n^2 <2+a_n$

    but $\displaystyle x^2<2+x$ for $\displaystyle x<2$

    so we must prove $\displaystyle a_n<2$ and then

    $\displaystyle a_n^2 <2a_n<a_n+2$ and so

    $\displaystyle a_n<\sqrt{2+a_n}=a_{n+1}$
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