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Math Help - Root test & Ratio test

  1. #1
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    Root test & Ratio test

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  2. #2
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    Ratio test : first of all, if \frac{a_{k+1}}{a_k} \rightarrow \infty, then the sequence does not converge, which means the series \sum a_k surely won't converge.

    Same with the root test - if limsup (a_k)^\frac{1}{k} \rightarrow \infty, then a_k \rightarrow \infty, and the sequence diverges, so the series will diverge too.
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  3. #3
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    Quote Originally Posted by adam63 View Post
    Ratio test : first of all, if \frac{a_{k+1}}{a_k} \rightarrow \infty, then the sequence does not converge, which means the series \sum a_k surely won't converge.

    Same with the root test - if limsup (a_k)^\frac{1}{k} \rightarrow \infty, then a_k \rightarrow \infty, and the sequence diverges, so the series will diverge too.
    So it looks like the root test and ratio test is applicable for the case c=∞ and r=∞ as well. (and we can say c=∞>1 or r=∞>1, hence the series diverges)

    But I don't understand why your claims are true. Can you explain/justify?

    Thank you!
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  4. #4
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    a series can converge ONLY if the sequence converges.
    Then, if the sequence diverges, the series will surely diverge .
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