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1+2+3+...+ n = n(n+1)/2
So 1/(1+2+3+...+ n) = 2/n(n+1) = 2/n - 2/(n+1)
For example,
1/(1+2+3) = 2/3 - 2/4 = 1/6
1/(1+2+3+4) = 2/4 - 2/5 = 1/10
So the sum is:
1 + (2/2-2/3) + (2/3-2/4) + (2/4-2/5) +... + (2/100 - 2/101)
= 1 + 1 - 2/101
= 200/101

1+2+3+ 4+5+6+ 7+8+ 9+ 10=55
+ + + + + + + + + +
10+9+8+7+6+5+4+3+2+ 1=55
= = = = = = = = = =
11 11 11 11 11 11 11 11 11 11
=110/2
=55
1+.....100
=(1+100)+(2+99)+(3+98)+(4+97)+(5+96)......(99+2)+(100+1)
=101+101+101+101+101+101+101+......101+101+101+101+101+101+
101+101+101
=101*100/2
=10100/2
=5050

2008-10-18 21:21:39 補充：
1+......+9+10=55
1+......+99+100=5050
1+......+999+1000=500500
1+......+9999+10000=50005000
1+......+99999+100000=5000050000

2008-10-18 21:21:44 補充：
1+......+999999+1000000=500000500000
1+......+9999999+10000000=50000005000000
1+......+99999999+100000000=5000000050000000
1+......+999999999+1000000000=500000000500000000
1+......+9999999999+10000000000=50000000005000000000