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Math Help - Series- Real Analysis

  1. #1
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    Question Series- Real Analysis

    Consider the series:

    1-1/2 -1/3 +1/4 +1/5 -1/6 -1/7........where the signs come in pairs. Does the series converge or diverge?

    I want to use the alternating series test....any ideas? how I would solve this.

    Thanks
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  2. #2
    Behold, the power of SARDINES!
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    Quote Originally Posted by hebby View Post
    Consider the series:

    1-1/2 -1/3 +1/4 +1/5 -1/6 -1/7........where the signs come in pairs. Does the series converge or diverge?

    I want to use the alternating series test....any ideas? how I would solve this.

    Thanks
    Try groupin together in pairs to get

    \left( \frac{2-1}{1\cdot2}\right) + \left( \frac{-4+3}{3\cdot 4}\right)+\left( \frac{6-5}{5\cdot 6}\right)+...+\frac{(-1)^{n+1}}{(2n-1)(2n)}+...
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  3. #3
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    Question

    but in your case we get a plus and then a minus.....but the series is minus minus ...plus plus?
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  4. #4
    Behold, the power of SARDINES!
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    Quote Originally Posted by hebby View Post
    but in your case we get a plus and then a minus.....but the series is minus minus ...plus plus?
    As I said add them to gether in pairs. Write it out and you will see.

    a_1+a_2=1-\frac{1}{2}=\frac{1}{2}=b_1
    a_3+a_4=-\frac{1}{3}+\frac{1}{4}=-\frac{1}{12}=b_2
    a_5+a_6=\frac{1}{5}-\frac{1}{6}=\frac{1}{30}=b_3

    Note this is the series listed above I just gave the general term.

    Since
    \sum a_n =\sum b_n we can look at the new series to draw any needed conclusions.
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