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Math Help - infinite series

  1. #1
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    infinite series

    Heys I'm having a few problems understanding a few worded questions and wonder if anyone may be able to help me.

    1. Suppose that an infinite series has the property that, given any positive number, all but
    fi nitely many terms of the series are positive and less than this number. Does it follow that
    this series converges?

    2. Can one ever obtain a convergent infinite series by interspersing the terms of two divergent
    series?

    3. Given any positive number, all but infinitely many partial sums of a certain infinite series aregreater than this number. Does it then follow that this series diverges?

    4. Can one determine whether a given infinite series converges or diverges merely by computing
    a suciently large number of partial sums?

    5. Can one determine the sum|accurate to a given fixed number of decimal places|of a
    convergent geometric series merely by computing a suciently large number of partial sums?

    Thanks in advance
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  2. #2
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    Re: infinite series

    Well, this is a math forum, not an English forum, where people could help you with understanding questions in the form of plain English sentences.

    First, it is not recommended to post more than two questions in one thread. Second, you need to describe more precisely what you don't understand.
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  3. #3
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    Re: infinite series

    Quote Originally Posted by Dragonkiller View Post
    Heys I'm having a few problems understanding a few worded questions and wonder if anyone may be able to help me.

    1. Suppose that an infinite series has the property that, given any positive number, all but
    finitely many terms of the series are positive and less than this number. Does it follow that
    this series converges?

    2. Can one ever obtain a convergent infinite series by interspersing the terms of two divergent
    series?

    3. Given any positive number, all but infinitely many partial sums of a certain infinite series aregreater than this number. Does it then follow that this series diverges?

    4. Can one determine whether a given infinite series converges or diverges merely by computing
    a suciently large number of partial sums?

    5. Can one determine the sum|accurate to a given fixed number of decimal places|of a
    convergent geometric series merely by computing a suciently large number of partial sums?

    Thanks in advance
    2, yes it is possible to get a convergent series by interspersing terms of two divergent series.

    \displaystyle \begin{align*} \sum_{n = 1}^{\infty}\frac{(-1)^{n + 1}}{n} = \log{2} \end{align*}, but we can also write

    \displaystyle \begin{align*} \sum_{n  = 1}^{\infty}\frac{(-1)^{n+1}}{n} &= 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \dots - \dots \\ &= \left(1 + \frac{1}{3} + \frac{1}{5} + \dots \right) - \left(\frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \dots \right) \\ &= \sum_{n = 0}^{\infty}\frac{1}{2n + 1} - \sum_{n = 1}^{\infty}\frac{1}{2n}  \end{align*}

    Each of these sums is divergent.
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  4. #4
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    Re: infinite series

    Quote Originally Posted by Dragonkiller View Post
    Heys I'm having a few problems understanding a few worded questions and wonder if anyone may be able to help me.

    1. Suppose that an infinite series has the property that, given any positive number, all but
    finitely many terms of the series are positive and less than this number. Does it follow that
    this series converges?

    2. Can one ever obtain a convergent infinite series by interspersing the terms of two divergent
    series?

    3. Given any positive number, all but infinitely many partial sums of a certain infinite series aregreater than this number. Does it then follow that this series diverges?

    4. Can one determine whether a given infinite series converges or diverges merely by computing
    a suciently large number of partial sums?

    5. Can one determine the sum|accurate to a given fixed number of decimal places|of a
    convergent geometric series merely by computing a suciently large number of partial sums?

    Thanks in advance
    For 4, the answer is... Sort of...

    Generally speaking, you would work out what expression you would get after you sum n terms, and then take the limit of this expression as n goes to infinity.
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  5. #5
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    Re: infinite series

    Quote Originally Posted by Dragonkiller View Post
    4. Can one determine whether a given infinite series converges or diverges merely by computing a suciently large number of partial sums?
    I understand this question in the following way. Can one determine if a series converges after numerically computing a finite number of partial sums?
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  6. #6
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    Re: infinite series

    Quote Originally Posted by emakarov View Post
    I understand this question in the following way. Can one determine if a series converges after numerically computing a finite number of partial sums?
    Yes, it all comes down to interpretation of the question, that's why I said "sort of", because doing this will give you an idea of what the limit will approach (if it does approach one).
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