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Math Help - Study the convergence of a sequence

  1. #1
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    Study the convergence of a sequence

    Hello all.

    How do you study the convergence of a sequence?
    If you can, try to exemplify with this example:
    U_n = 5 - \frac{2}{\sqrt[3]{n}}

    Thanks in advance!
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  2. #2
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    Have a read of this.

    Ratio test - Wikipedia, the free encyclopedia

    Your sequence looks like it converges to 5.
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  3. #3
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    Quote Originally Posted by Wright View Post
    Hello all.

    How do you study the convergence of a sequence?
    If you can, try to exemplify with this example:
    U_n = 5 - \frac{2}{\sqrt[3]{n}}

    Thanks in advance!

    Arithmetic of limits: \frac{2}{\sqrt[3]{n}}\xrightarrow [n\to \infty] {} 0\,,\,\,and\,\,5\xrightarrow [n\to \infty] {} 5, so the limit of both summands exists and finite and thus the limit is the sum of these two limits, i.e. 5.

    Tonio
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  4. #4
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    Quote Originally Posted by pickslides View Post
    Have a read of this.

    Ratio test - Wikipedia, the free encyclopedia

    Your sequence looks like it converges to 5.
    Be careful.The ratio is test can be used only for series.

    The only way you can relate the two is that if you show

    \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|<  1

    you know the series \sum_{n\in\mathbb{N}}a_n converges which means that \lim_{n\to\infty}a_n=0 But that is not applicable here
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  5. #5
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    So it's convergent if theres a limit and divergent if there isn't one?
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  6. #6
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    Correct.
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  7. #7
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    Limited sequence?

    Mistake post.
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