2=1
Sometimes seemingly logical arguments deceive you.
S = 1 – ½ + 1/3 – ¼ + 1/5 – 1/6 + … = ½ + 1/3 – ¼ + 1/5 – 1/6 + …
From left hand side, it can be shown that S<1, from the right hand side S>½ (how?).
Multiply the series by 2.
2S = 2 – 1 + 2/3 – ½ + 2/5 – 1/3 + …
Rearrange the terms:
2S = (2 – 1 ) – ½ + (2/3 – 1/3) – ¼ + (2/5 – 1/5) - …
= 1 – ½ + 1/3 – ¼ + 1/5 - … = S
Since S converges and is not zero, divide both side by S and arrive at 2=1.
posted by Mehrdad at
7:06 PM
2 Comments:
You can't simply subract the right side of 1+(2/3)-(1/2)... from only the middle term 2-(1/2)...
By
MECU, at 7:26 PM
the terms of 2S are just re-arranged and re-grouped. All the terms are preserved...
By
Mehrdad, at 7:36 PM
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