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Thread: Does this limit proof work?

  1. #1
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    Does this limit proof work?

    I have to prove that $\displaystyle a_{n}=\frac {1+2+3+...+n}{n^{2}}$ converges to 1/2.

    Proof: We want to show that $\displaystyle a_{n}$ converges to 1/2. So we want to show that
    $\displaystyle \left|\frac {1+2+3+...+n}{n^{2}}-\frac{1}{2}\right|<\epsilon$
    Note that $\displaystyle a_{n}'=\frac {1+2+3+...+n}{n}>\frac {1+2+3+...+n}{n^{2}}=a_{n}$.
    Thus if we prove the condition for $\displaystyle a_{n}'$ then we simultaneously prove it for $\displaystyle a_{n}$.
    So $\displaystyle a_{n}'-\frac{1}{2}=\frac{n(n+1)}{2n}-\frac{1}{2}=\frac{n(n+1)-n}{2n}=\frac{n}{2}$.
    So take $\displaystyle \frac{n}{2}<\epsilon \implies n<2\epsilon$.
    So choose $\displaystyle n^{*}<n<2\epsilon$, and the condition holds for $\displaystyle a_{n}' \implies$ the condition holds for $\displaystyle a_{n}$ since $\displaystyle a_{n}<a_{n}'$. QED.

    Is this a valid proof?
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  2. #2
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    Why not use $\displaystyle \displaystyle\sum_{k=1}^{n} k = \dfrac{n(n+1)}{2}$?
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  3. #3
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by paupsers View Post
    I have to prove that $\displaystyle a_{n}=\frac {1+2+3+...+n}{n^{2}}$ converges to 1/2.

    Proof: We want to show that $\displaystyle a_{n}$ converges to 1/2. So we want to show that
    $\displaystyle \left|\frac {1+2+3+...+n}{n^{2}}-\frac{1}{2}\right|<\epsilon$
    Note that $\displaystyle a_{n}'=\frac {1+2+3+...+n}{n}>\frac {1+2+3+...+n}{n^{2}}=a_{n}$.
    Thus if we prove the condition for $\displaystyle a_{n}'$ then we simultaneously prove it for $\displaystyle a_{n}$. This is not true.
    So $\displaystyle a_{n}'-\frac{1}{2}=\frac{n(n+1)}{2n}-\frac{1}{2}=\frac{n(n+1)-n}{2n}=\frac{n}{2}$.
    So take $\displaystyle \frac{n}{2}<\epsilon \implies n<2\epsilon$. You want n to be greater than some function of epsilon, not less than it.
    So choose $\displaystyle n^{*}<n<2\epsilon$, and the condition holds for $\displaystyle a_{n}' \implies$ the condition holds for $\displaystyle a_{n}$ since $\displaystyle a_{n}<a_{n}'$. QED.

    Is this a valid proof?
    Proof:

    $\displaystyle \left|\frac{\sum_{k=1}^n k}{n^2}-\frac{1}{2}\right|<\epsilon$

    $\displaystyle \left|\frac{\frac{n(n+1)}{2}}{n^2}-\frac{1}{2}\right|<\epsilon$

    $\displaystyle \left|\frac{n+1}{2n}-\frac{1}{2}\right|<\epsilon$

    $\displaystyle \left|\frac{n+1-n}{2n}\right|<\epsilon$

    $\displaystyle \left|\frac{1}{2n}\right|<\epsilon$

    So for any $\displaystyle \epsilon>0$, let $\displaystyle n>\frac{1}{2\epsilon}$, and the condition will hold. $\displaystyle \square$
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