數學知識交流-Compair

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

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 larger