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Math Help - difference of a diverging series

  1. #1
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    difference of a diverging series

    Hello,
    If \sum_{1}^{\infty} a_n is a diverging series, is \sum_{1}^{\infty} a_n -  \sum_{1}^{\infty} a_n equal to zero, or is it undefined?
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  2. #2
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    Since they are exactly the same in this case it would equal 0.

    If however, it was the difference of two different diverging series, it would be indeterminate.
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  3. #3
    MHF Contributor chisigma's Avatar
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    Quote Originally Posted by dudyu View Post
    Hello,
    If \sum_{1}^{\infty} a_n is a diverging series, is \sum_{1}^{\infty} a_n -  \sum_{1}^{\infty} a_n equal to zero, or is it undefined?
    That is a case where You must be carefull!... if \displaystyle \sum_{n=1}^{\infty} a_{n} diverges, the following expressions are not the same...

    a) \displaystyle \sum_{n=1}^{\infty} a_{n} - \displaystyle \sum_{n=1}^{\infty} a_{n}

    b) \displaystyle \sum_{n=1}^{\infty} \{a_{n} - a_{n}\}

    In a) each term of the 'sum' is a divergent series and the 'sum' is impossible. In b) You have a single series with all terms equal to 0, so that it converges and its sum is 0...

    Kind regards

    \chi \sigma
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