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Math Help - Proving the Ratio Test

  1. #1
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    Proving the Ratio Test

    The question I've been given involves proving two of the three different scenarios of the Ratio Test. We've been given it as follows.

    Let \sum a_k be a series with positive terms and suppose that \frac{a_{k+1}}{a_k}\rightarrow\lambda.
    1. If \lambda<1, then \sum a_k converges. (this proof was given to us already)
    2. If \lambda>1, then \sum a_k diverges.
    3. If \lambda=1, then the test is inconclusive.

    The second and third parts need to be proven.

    I believe I've gotten a good proof for the second one, where \sum a_k diverges, though I could use a second opinion.

    The third one, where the test is inconclusive, however, escapes me in terms of a proof. I could surely use a hand with that one.
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  2. #2
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    2. If \lim_{n \to \infty} \, \frac{a_{n+1}}{a_n} = \lambda > 1, then a_n \le a_{n+1} for large n. Therefore, \lim_{n \to \infty} \,a_n \not= 0 \Longrightarrow \sum a_n is divergent.

    3. Take the p-series \sum n^{-p}

    \lim_{n \to \infty} \, \frac{a_{n+1}}{a_n}=1 for any p.
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  3. #3
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    Quote Originally Posted by Runty View Post
    The question I've been given involves proving two of the three different scenarios of the Ratio Test. We've been given it as follows.

    Let \sum a_k be a series with positive terms and suppose that \frac{a_{k+1}}{a_k}\rightarrow\lambda.
    1. If \lambda<1, then \sum a_k converges. (this proof was given to us already)
    2. If \lambda>1, then \sum a_k diverges.
    3. If \lambda=1, then the test is inconclusive.

    The second and third parts need to be proven.

    I believe I've gotten a good proof for the second one, where \sum a_k diverges, though I could use a second opinion.

    The third one, where the test is inconclusive, however, escapes me in terms of a proof. I could surely use a hand with that one.
    For part three simply note that both series \sum\limits_{n = 1}^\infty  {\frac{1}{n}} \,\& \,\sum\limits_{n = 1}^\infty  {\frac{1}{{n^2 }}} yield the same result from the ratio test.
    But one diverges while the other converges.
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