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Grandi’s series
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
