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Thread: Sequences and their Limits

  1. #1
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    Sequences and their Limits

    Use Theorem 3 to show that the sequence 0.7, 0.77, 0.777, 0.7777, 0.77777, 0.777777, 0.7777777, ... has a limit.

    Theorem 3 is ...

    Suppose that the sequence $\displaystyle a_{k}$ is nondecresing and bounded above by a number A. That is,

    $\displaystyle a_{1} \leq a_{2} \leq a_{3} \leq A$

    Then $\displaystyle a_{k}$ converges to some finite limit a, with a $\displaystyle \leq A$. Similarly, if $\displaystyle b_{k}$ is nonincreasing and bounded below by a number B, then $\displaystyle b_{k}$ converges to a finite limit $\displaystyle b \geq B$.

    edit: How do I do the less than or equal to sign in LaTeX?
    Last edited by larson; Apr 20th 2008 at 06:51 PM.
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    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by larson View Post
    Use Theorem 3 to show that the sequence 0.7, 0.77, 0.777, 0.7777, 0.77777, 0.777777, 0.7777777, ... has a limit.

    Theorem 3 is ...

    Suppose that the sequence $\displaystyle a_{k}$ is nondecresing and bounded above by a number A. That is,

    $\displaystyle a_{1}$ <, $\displaystyle a_{2}$ < $\displaystyle a_{3}$ < A

    Then $\displaystyle a_{k}$ converges to some finite limit a, with a < A. Similarly, if $\displaystyle b_{k}$ is nonincreasing and bounded below by a number B, then $\displaystyle b_{k}$ converges to a finite limit b > B.

    edit: How do I do the less than or equal to sign in LaTeX?
    $\displaystyle /leq$ gives $\displaystyle \leq$ just change / to \
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  3. #3
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by larson View Post
    Use Theorem 3 to show that the sequence 0.7, 0.77, 0.777, 0.7777, 0.77777, 0.777777, 0.7777777, ... has a limit.

    Theorem 3 is ...

    Suppose that the sequence $\displaystyle a_{k}$ is nondecresing and bounded above by a number A. That is,

    $\displaystyle a_{1}$ <, $\displaystyle a_{2}$ < $\displaystyle a_{3}$ < A

    Then $\displaystyle a_{k}$ converges to some finite limit a, with a < A. Similarly, if $\displaystyle b_{k}$ is nonincreasing and bounded below by a number B, then $\displaystyle b_{k}$ converges to a finite limit b > B.

    edit: How do I do the less than or equal to sign in LaTeX?
    you can use the first part of the theorem with $\displaystyle A = 0. \bar{7}$ or $\displaystyle A = 0.8$
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  4. #4
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by larson View Post
    Use Theorem 3 to show that the sequence 0.7, 0.77, 0.777, 0.7777, 0.77777, 0.777777, 0.7777777, ... has a limit.

    Theorem 3 is ...

    Suppose that the sequence $\displaystyle a_{k}$ is nondecresing and bounded above by a number A. That is,

    $\displaystyle a_{1}$ <, $\displaystyle a_{2}$ < $\displaystyle a_{3}$ < A

    Then $\displaystyle a_{k}$ converges to some finite limit a, with a < A. Similarly, if $\displaystyle b_{k}$ is nonincreasing and bounded below by a number B, then $\displaystyle b_{k}$ converges to a finite limit b > B.

    edit: How do I do the less than or equal to sign in LaTeX?
    and I will give you a hint $\displaystyle a_n=7\sum_{t=1}^{n}\cdot\bigg(\frac{1}{10}\bigg)^{ t}$
    Last edited by Mathstud28; Apr 20th 2008 at 05:45 PM.
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  5. #5
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Mathstud28 View Post
    $\displaystyle /leq$ gives $\displaystyle \leq$ just change / to \
    it suffices to type le, likewise, ge and ne for greater thank and not equal to respectively
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