Are the Results of Adding and Subtracting ζ(0) and Grandi's Series Valid?

[Kekasi]

Recently I came across this information:

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

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

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

\text{S1}+\text{S2} = 2+0+2+0+\dotsb=\hspace{3.8mm}0
\text{S1}-\text{S2} = 0+2+0+2+\dotsb=-1My questions:
Are S1+S2 and S1-S2 valid? If not so, why?
If so, is the following true? If not so, why?

\text{S1}-\text{S2} = 0+2+0+2+0+\dotsb
\hspace{17.25mm}=\hspace{8.3mm}2+0+2+0+\dotsb=\text{S1}+\text{S2}

 

[mathman]

Recently I came across this information:

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

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

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

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


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

\text{S1}-\text{S2} = 0+2+0+2+0+\dotsb
\hspace{17.25mm}=\hspace{8.3mm}2+0+2+0+\dotsb=\text{S1}+\text{S2}

These are all mathematical fallacies.