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Math Help - Series

  1. #1
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    Series

    I have to answer the following true or false questions. But also be able to explain why. I think I have the first 2 right saying they are both true. But I can't really explain why and I don't know about the last 2.

    True or False

    a) If a series does not converge, then its nth term does not approach 0 as n approaches infinity.

    b) If the nth term of a series does not approach 0 as n approaches infinity, then the series diverges.

    c) If all partial sums of a series are less than some constant L, then the series converges.

    d) If a series converges, then there is a constant L such that all of its partial sums are less than L.
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  2. #2
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    Hello, TreeMoney!

    Here are a few of them . . .


    True or False

    a) If a series does not converge, then its nth term does not approach 0
    as n approaches infinity.

    False

    The Harmonic Series: 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \hdots does not converge

    but its n^{th} term does approach 0: . \lim_{n\to\infty}\frac{1}{n} \:=\:0



    b) If the nth term of a series does not approach 0 as n approaches infinity,
    then the series diverges.

    True

    A necessary condition for convergence is: . \lim_{n\to\infty} a_n \:=\:0



    c) If all partial sums of a series are less than some constant L,
    then the series converges.

    False

    The oscillating series: .  1 - 1 + 1 - 1 + \hdots has a sum less than 2,
    . . but it diverges.



    d) If a series converges, then there is a constant L
    such that all of its partial sums are less than L.

    I'll let you think about this one . . .

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  3. #3
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    Quote Originally Posted by TreeMoney View Post

    d) If a series converges, then there is a constant L such that all of its partial sums are less than L.
    Yes, ever convergent sequence (series is a type of sequence) is bounded.
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