- #1
Kekasi
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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]
[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: