非常妙的解答[From昌爸]:
a=1/(1*2)+1/(3*4)+...+1/(2003*2004) =1-1/2+1/3-1/4+...+1/2003-1/2004 =1+1/2+1/3+...+1/2004-2(1/2+1/4+...+1/2004) =1+1/2+1/3+...+1/2004-(1+1/2+...+1/1002) =1/1003+1/1004+...+1/2004
b=1/(1003*2004)+1/(1004*2003)+...+1/(2003*1004)+1/(2004*1003) =2[1/(1003*2004)+1/(1004*2003)+...+1/(1503*1504)] =(2/3007)(1/1003+1/2004+1/1004+1/2003+...+1/1503+1/1504) =(2/3007)(1/1003+1/1004+...+1/2004)
所以 a/b=3007/2
_________________ 想要得到什麼,就必須付出相同的代價...這就是煉金術中所說的"等價交換原則".
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