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What is the sum of all integers?

 Jun 27, 2020
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A very interesting question ...

 

The integers:   ..., -3, -2, -1, 0, 1, 2, 3, ...

 

If you arranged them as:  0, 1, -1, 2, -2, 3, -3, 4, -4, ...

 

and added them this way:  (0) + (1 + -1) + (2 + -2) + (3 + -3) + ...

       you would get:             0 + 0 + 0 + 0 + ...  =  0

 

but, if you added them this way:  (0 + 1) + (-1 + 2) + (-2 + 3) + (-3 + 4) + ...

       you would get:                       1 + 1 + 1 + 1 + 1 + ...  which tends to positive infinity

 

and, if you added them this way:  (0 + -1) + (1 + -2) + (2 + -3) + (3 + -4) + ...

       you would get:                        -1 + -1 + -1 + -1 + ...  which tends to negative infinity

 

So, there is no particular sum  --  no sum ...

 

You might want to try to rearrange the integers to get other "answers" ...

 Jun 27, 2020

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