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Math Help - Quick question on series

  1. #1
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    Quick question on series

    If I know that a series
    <br />
\sum a_n <br />
    is convergent. Is it not possible to further deduce that
    <br />
\lim\limits_{n\to\infty} \frac{a_{n+1}}{a_n} \leq 1<br />

    Hjörtur
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  2. #2
    Super Member Matt Westwood's Avatar
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    Quote Originally Posted by hjortur View Post
    If I know that a series
    <br />
\sum a_n <br />
    is convergent. Is it not possible to further deduce that
    <br />
\lim\limits_{n\to\infty} \frac{a_{n+1}}{a_n} \leq 1<br />

    Hjörtur
    The implication in the other direction is straightforward: it's the Ratio Test:

    Ratio Test - ProofWiki

    Does that help at all?
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  3. #3
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    I just wanted to be sure, I am pretty sure that the limit must hold.
    What if a_n=0 for all n. What happens to the limit then? Undefined?
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  4. #4
    Super Member Matt Westwood's Avatar
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    Quote Originally Posted by hjortur View Post
    I just wanted to be sure, I am pretty sure that the limit must hold.
    What if a_n=0 for all n. What happens to the limit then? Undefined?
    Looks like it to me. The ratio test definitely holds in the other direction (as I've said) but you've found a counterexample such that the way you stated it does *not* hold.
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  5. #5
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    Then i'll just assume that particular series is not 0. Thanks
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