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richyw

limits. How do I show that \[\lim_{k\to\infty} \frac{(-1)^k}{k}=0\]

  • one year ago
  • one year ago

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  1. shubhamsrg
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    Do you know LH rule ?? Where you can differentiate the numerator and denominator.

    • one year ago
  2. richyw
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    sorry I keep messing up the latex and openstudy is so slow.

    • one year ago
  3. richyw
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    and yeah I know lh rule could be applied since \(k\in\mathbb{R}\). But how does that help?

    • one year ago
  4. shubhamsrg
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    Okay since it is (-1) , LH rule won't be applied.

    • one year ago
  5. shubhamsrg
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    hmm, so, think like this : (-1)^n , where n is any natural number, will always be equal to +1 or -1 right ?

    • one year ago
  6. richyw
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    agreed. and since \(k\to\infty\) I can see that it's 0. just not sure how to show that in any sort of rigorous way.

    • one year ago
  7. richyw
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    I need to show it though to show that a series is convergent.

    • one year ago
  8. shubhamsrg
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    hmm, Am not too comfortable with that. @phi

    • one year ago
  9. phi
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    I thought the definition of a limit L is that for any \( \epsilon >0\), there exists an integer N>0 such that \( | L - x_n| < \epsilon \) for all n>N the alternating sign does not affect this definition, because you are only looking at the distance away from L. (0 in this case)

    • one year ago
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