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數學知識交流-Compair

發問:

1) Which is bigger ? A.1222...2223(102 units) * 2333...3334(102 units) B.4555...5556(102 units) * 676767...676(101 units) 2) Which is smaller ? A.1/2+1/3+1/4+1/5+...+1/98+1/99+1/100 B.1/101+1/102+1/103+...+1/99999+1/100000 更新: Compare

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最佳解答:

1)A = 1222...2223 x 2333...3334 < 1230...0000 x 2340...0000 = (123 x 234) x (10^198) = 28782 x (10^198)B= 4555...5556 x 6767...676 > 4550...0000 x 6760...000 = (455 x 67.6) x (10^198) = 30758 x (10^198)故 A < 28782 x (10^198) < 30758 x (10^198) < B 2)A = 1/2 + 1/3 + 1/4 + 1/5 +...+ 1/98 + 1/99 + 1/100= 1000(1/2,000) .........line 1 + 1000(1/3,000) .........line 2 + 1000(1/4,000) .........line 3 + ....... + 1000(1/99,999) .......line 98 + 1000(1/100,000) .....line 99< B = 1/101 + 1/102 + 1/103 +...+ 1/99999 + 1/100000= (1/101 + ... + 1/1100) .........line 1'' > line 1 + (1/1,101 + ... + 1/2100) .......line 2'' > line 2 + (1/2,101 + ... + 1/3100) ..........line 3'' > line 3 + ....... + (1/98,001 + ... + 1/99000) ...........line 99'' > line 99 + (1/99,001 + ... + 1/100000) ...........line 100Therefore A < B . 2011-06-06 21:24:45 補充： line 98 should be 1/99,000 2011-06-06 22:23:50 補充： 修正最後三行 : + (1/98,101 + ... + 1/99,100) ...........line 99'' > line 99 + (1/99,101 + ... + 1/100,00) ...........line 100 Therefore A < B . 2011-06-06 22:25:46 補充： + (1/99,101 + ... + 1/100,000) ...........line 100

其他解答:

1) Which is bigger ? B.4555...5556(102 units) * 676767...676(101 units) 2) Which is smaller ? B.1/101+1/102+1/103+...+1/99999+1/100000|||||For #1.) Since 12*23 < 45*6.7 Hence, B is largerA215E4A2B88AAE64