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Thread: Limit of a recursively defined sequence

  1. #1
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    Limit of a recursively defined sequence

    I'm having some trouble with this question.

    Suppose that $\displaystyle a_1=1$ and $\displaystyle a_{n+1}=\sqrt{1+a_n}$ whenever $\displaystyle n\geq 1$

    1) Show that $\displaystyle \sqrt{1+x}\in [1,2]$ whenever $\displaystyle x\in [1,2]$.


    2) Use induction to show that the sequence $\displaystyle \{a_n\}$ is bounded.

    Would I do the question like this:

    Prove that $\displaystyle a_n<2 \ \ \ \forall n\geq 1$

    $\displaystyle a_1=1<2$

    Assume $\displaystyle a_k<2$, then $\displaystyle a_{k+1}=\sqrt{1+a_k}<\sqrt{3}<2$

    By induction it follows that $\displaystyle a_n<2 \ \ \ \forall n\geq 1 \Rightarrow a_n$ is bounded above by 2.



    3) Use induction to show that $\displaystyle {a_n}$ is an increasing sequence

    Prove $\displaystyle a_{n+1}>a_n \ \ \ \forall n\geq 1$

    $\displaystyle a_2=\sqrt{1+1}=\sqrt{2}>a_1=1$

    Assume that $\displaystyle a_{k+1}>a_k$

    $\displaystyle a_{k+2}=\sqrt{1+a_{k+1}}>\sqrt{1+a_k}=a_{k+1}$

    By induction it follows that, $\displaystyle a_{n+1}>a_n \ \ \ \forall n\geq 1 \Rightarrow a_n$ is an increasing sequence



    4) Explain why $\displaystyle \lim_{x \to \infty}a_n$ exists

    Since $\displaystyle a_n$ is a monotonically increasing sequence, bounded above, it must converge to a limit as $\displaystyle n\rightarrow\infty$



    5) Find $\displaystyle \lim_{x \to \infty}a_n$. (That is, find $\displaystyle \sqrt{1+\sqrt{1+\sqrt{1+...}}}$

    Can someone check whether what I did is correct and help me with (1) and (5)
    Last edited by acevipa; Sep 13th 2010 at 03:03 AM.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by acevipa View Post
    I'm having some trouble with this question.

    Suppose that $\displaystyle a_1=1$ and $\displaystyle a_{n+1}=\sqrt{1+a_n}$ whenever $\displaystyle n\geq 1$

    1) Show that $\displaystyle \sqrt{1+x}\in [1,2]$ whenever $\displaystyle x\in [1,2]$.

    Can someone check whether what I did is correct and help me with (1) and (5)
    $\displaystyle f(x)=\sqrt{1+x}$ is increasing for $\displaystyle x \in [0,\infty)$ , so we hace:

    $\displaystyle 1\le f(x)\le \sqrt{3}<2$ for $\displaystyle x\in [1,2]$

    CB
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  3. #3
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    Quote Originally Posted by acevipa View Post
    5) Find $\displaystyle \lim_{x \to \infty}a_n$. (That is, find $\displaystyle \sqrt{1+\sqrt{1+\sqrt{1+...}}}$

    Can someone check whether what I did is correct and help me with (1) and (5)
    You have shown that:

    $\displaystyle a=\lim_{x \to \infty}a_n$

    exists, and so is a solution of $\displaystyle a=\sqrt{1+a}$

    CB
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  4. #4
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    Quote Originally Posted by CaptainBlack View Post
    You have shown that:

    $\displaystyle a=\lim_{x \to \infty}a_n$

    exists, and so is a solution of $\displaystyle a=\sqrt{1+a}$

    CB
    I managed to work out (5)

    $\displaystyle \lim_{n \to \infty}a_{n+1}=\lim_{n \to \infty}\sqrt{1+a_n}\Rightarrow a=\sqrt{1+a}\Rightarrow a^2=1+a$

    $\displaystyle \Rightarrow a^2-a-1=0\Rightarrow a=\frac{1\pm\sqrt{1+4}}{2}$

    But $\displaystyle a_n>0$, so $\displaystyle \lim_{n\to\infty}a_n=\frac{1+\sqrt{5}}{2}$

    By the way, is my question 2, 3, 4 correct?
    Last edited by acevipa; Sep 13th 2010 at 03:38 AM.
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  5. #5
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    Quote Originally Posted by acevipa View Post
    I don't quite understand what you did.

    The answer in the back of the book is $\displaystyle \frac{1+\sqrt{5}}{2}$

    By the way, is my question 2, 3, 4 correct?
    2, 3 and 4 look OK, but could use some tidying up.

    what I did was to observe that:

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

    implies that (since the limit exists):

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

    or:

    $\displaystyle a=\sqrt{1+a}$

    CB
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