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Math Help - Give examples of sequences;Determine which of the following series converges and comp

  1. #1
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    Give examples of sequences;Determine which of the following series converges and comp

    A)
    Give examples of sequences:
    a) one that converges to 100
    b) one that converges to 3/4
    c) one that diverges

    Justify what you are presenting.

    B)
    Determine which of the following series converges and compute its sum:
    a) n=0, infinite 1/n^(1/3)
    b) n=0, infinite 4^n

    Justify your answer.
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  2. #2
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    Quote Originally Posted by ewkimchi View Post
    A)
    Give examples of sequences:
    a) one that converges to 100
    b) one that converges to 3/4
    c) one that diverges

    Justify what you are presenting.

    B)
    Determine which of the following series converges and compute its sum:
    a) n=0, infinite 1/n^(1/3)
    b) n=0, infinite 4^n

    Justify your answer.
    What have you tried here? Where are you stuck?

    For A), note that 1/n --> 0 ....

    For B), neither series converges. Your job now is to think about why.
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  3. #3
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    Did I get it right?

    Do these work?

    A)
    a) (99n + 1)/100n

    b) (3n+1)/4n

    c) (n^3)/(n^2 + 1)

    B)
    a)
    by p series test, diverges b/c 1/3<1

    b)
    by geometric series test, converges b/c 1/4<1 and sum=8/3

    c)
    I'm not sure about this one because 4 isn't under a fraction or does that even matter?
    Anyway, I got
    by geometric series test diverges b/c 4>1
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  4. #4
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    Quote Originally Posted by ewkimchi View Post
    Do these work?

    A)
    a) (99n + 1)/100n
    No, this one converges to \color{blue} \frac{99}{100}.
    b) (3n+1)/4n
    Correct.
    c) (n^3)/(n^2 + 1)
    Correct.

    B)
    a)
    by p series test, diverges b/c 1/3<1
    Correct.
    b)
    by geometric series test, converges b/c 1/4<1 and sum=8/3
    The series in your question is 4^n not (1/4)^n !
    c)
    I'm not sure about this one because 4 isn't under a fraction or does that even matter?
    Anyway, I got
    by geometric series test diverges b/c 4>1
    There is no part c in your question !
    And remember that, you should check the absolute value of the common ratio.
    For instance,
    \sum_{n=0}^{\infty} \left(\frac{-3}{2}\right)^n DIVERGES.
    because |-3/2|=3/2>1.
    Anyway:

    For \color{blue} \sum_{n=0}^{\infty} 4^n : The series diverges by the nth term test(Test for Divergence).

    For \color{blue} \sum_{n=0}^{\infty} \left(\frac{1}{4}\right)^n : The series converges b/c its a geomreic series with common ration |1/4|=1/4< 1.
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