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Grandi’s series

Grandi’s series

Consider 1 – 1 + 1 – 1 + ….

Is the answer 0, 1 or ½? Or all three?

Consider 1 – 1 + 1 – 1 + ….

Is the answer 0, 1 or ½? Or all three?

Approaches:

[A]    (1 – 1) + (1 – 1) + (1 – 1) + … = 0 + 0 + 0 + … = 0

 

[B]     1 + (– 1 + 1) + (– 1 + 1 ) + (– 1 + 1) + … = 1 + 0 + 0 + 0 + … = 1

 

[C]     S = (1 – 1 + 1 – 1 + 1 – 1 +  …) , so

    1 – S = 1 – (1 – 1 + 1 – 1 + 1 – 1 + …) = 1 – 1 + 1 – 1 + 1 – 1 + 1 … = S

    1 – S = S

          1 = 2S

          S = ½

Yes

Yes