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Sum of two divergent series

  1. Sep 12, 2015 #1
    Consider the two divergent series:
    $$\sum_{n=k}^{\infty} a_n$$
    $$\sum_{n=k}^{\infty} b_n$$
    Is it possible for ##\sum_{n=k}^{\infty} (a_n \pm b_n)## to converge?
     
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  3. Sep 12, 2015 #2

    Orodruin

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    Yes, consider the case ##a_n = b_n##. Then the sum of ##a_n - b_n## is zero.
     
  4. Sep 12, 2015 #3
    Thanks!
    But in that case does it make sense to say that ##\sum_{n=k}^{\infty} (a_n \pm b_n) = \sum_{n=k}^{\infty} a_n \pm \sum_{n=k}^{\infty} b_n##?
     
  5. Sep 12, 2015 #4

    Orodruin

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    No, it generally make sense only if the series converge.
     
  6. Sep 12, 2015 #5

    PeroK

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    You really ought to be able to find examples yourself to resolve this question.
     
  7. Sep 13, 2015 #6

    Ssnow

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    You can construct others, for example ##a_ {n}=\frac{1}{n^{2}}+b_{n}## so ## \sum_{n=1}^{\infty}a_{n}-b_{n}=\frac{\pi^{2}}{6}##...
     
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