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Math Help - series questions, not a problem.

  1. #1
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    series questions, not a problem.

    Are the geometric, harmonic, and collapsing series all the same?

    Or can i only test the geometric series by this formula: a/(1-r)

    is it that the harmonic series ALWAYS diverges.
    Is this because it continues on infinitly?

    and the collapsing series, In order to determine if it converges, must i first put the equation in a two fractions form, through partial fraction substitution for example?

    If i need to show specific problem examples i can, any further explanation to my questions can help. Just using basic calculus 3 skills here. thankyou.
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  2. #2
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    Quote Originally Posted by rcmango View Post
    Are the geometric, harmonic, and collapsing series all the same?
    .
    I do not know what you mean. They are all three different series.
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  3. #3
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    how about the other questions please.

    Or can i only test the geometric series by this formula: a/(1-r)

    is it that the harmonic series ALWAYS diverges.
    Is this because it continues on infinitly?

    and the collapsing series, In order to determine if it converges, must i first put the equation in a two fractions form, through partial fraction substitution for example?

    If i need to show specific problem examples i can, any further explanation to my questions can help. Just using basic calculus 3 skills here. thankyou.
    In other words, how do i know the difference between a harmonic series and a geometric series?
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  4. #4
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    Quote Originally Posted by rcmango View Post
    Are the geometric, harmonic, and collapsing series all the same?

    Or can i only test the geometric series by this formula: a/(1-r)

    is it that the harmonic series ALWAYS diverges.
    Is this because it continues on infinitly?

    and the collapsing series, In order to determine if it converges, must i first put the equation in a two fractions form, through partial fraction substitution for example?

    If i need to show specific problem examples i can, any further explanation to my questions can help. Just using basic calculus 3 skills here. thankyou.
    Harmonic Series:

    \sum_{k=1}^{\infty}\frac{1}{k}

    diverges. It diverges because the terms do not decrease fast enough for
    the sum to converge. The terms can be grouped into ever larger groups each
    of which has a sum greater than (or equal) to 1/2, and because there are still
    an infinite number of such groups the sum diverges (for more details see either
    the wikipedia or mathworld pages on the harmonic series)

    The n-th parial sum of the harmonic series:

    H_n=\sum_{k=1}^{n}\frac{1}{k}

    is the n-th harmonic number

    RonL
    Last edited by CaptainBlack; January 26th 2007 at 03:46 AM.
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  5. #5
    Grand Panjandrum
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    Quote Originally Posted by rcmango View Post
    Are the geometric, harmonic, and collapsing series all the same?
    Collapsing series? Is this the same as a telescoping series?

    A telescoping series is one in which the general term a_n may
    be written as the difference of consecutive terms of a sequence, that is:

    a_n=b_n-b_{n+1}.

    Then:

    \sum_1^N a_n = \sum_{n=1}^N (b_n-b_{n+1})

    but now we have cancellation of all the b's except the first and last, so:

    \sum_1^N a_n = \sum_{n=1}^N (b_n-b_{n+1})=b_1-b_{N+1}

    RonL
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