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Ζ(0) and Grandi's Series

  1. Aug 12, 2014 #1
    Recently I came across this information:

    [itex]\text{S1} = 1+1+1+1+\dotsb= -\frac{1}{2}[/itex]
    [itex]\text{S2} = 1-1+1-1+\dotsb= \hspace{4.6mm} \frac{1}{2}[/itex]

    Which are ζ(0) and Grandi's series respectively.

    After tinkering with this information, I produced these strange results.

    [itex]\text{S1}+\text{S2} = 2+0+2+0+\dotsb=\hspace{3.8mm}0[/itex]
    [itex]\text{S1}-\text{S2} = 0+2+0+2+\dotsb=-1[/itex]


    My questions:
    Are S1+S2 and S1-S2 valid? If not so, why?
    If so, is the following true? If not so, why?

    [itex]\text{S1}-\text{S2} = 0+2+0+2+0+\dotsb[/itex]
    [itex]\hspace{17.25mm}=\hspace{8.3mm}2+0+2+0+\dotsb=\text{S1}+\text{S2}[/itex]
     
    Last edited: Aug 12, 2014
  2. jcsd
  3. Aug 13, 2014 #2

    mathman

    User Avatar
    Science Advisor
    Gold Member

    These are all mathematical fallacies.
     
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