Saturday, November 13, 2004
2 = 1
گول ظاهر بعضی چیزها را نباید خورد، حتی اگر در نگاه اول منطقی به نظر بیاید.
Sometimes seemingly logical arguments deceive you.
Consider the series
S = 1 – ½ + 1/3 – ¼ + 1/5 – 1/6 + … = ½ + 1/3 – ¼ + 1/5 – 1/6 + … x
From left hand side, it can be shown that S < 1, and from the right hand side, S > ½ (how?) . Multiply the series by 2.
2S = 2 – 1 + 2/3 – ½ + 2/5 – 1/3 + … x
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 1/2 < S < 1, we can divide both sides by S, and arrive at 2 = 1
S = 1 – ½ + 1/3 – ¼ + 1/5 – 1/6 + … = ½ + 1/3 – ¼ + 1/5 – 1/6 + … x
From left hand side, it can be shown that S < 1, and from the right hand side, S > ½ (how?) . Multiply the series by 2.
2S = 2 – 1 + 2/3 – ½ + 2/5 – 1/3 + … x
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 1/2 < S < 1, we can divide both sides by S, and arrive at 2 = 1
