比較
1 - 1/2 + 1/3 - 1/4 + ... + 1/2009 - 1/2010
及
1/1006 + 1/1007 + ... + 1/2009 + 1/2010
之大小。
小學數學 : 數式比大小
2010-12-12 11:15 pm
回答 (3)
2010-12-12 11:58 pm
✔ 最佳答案
1 - 1/2 + 1/3 - 1/4 + ... + 1/2009 - 1/2010 <1>1/1006 + 1/1007 + ... + 1/2009 + 1/2010 <2>
<2> - <1>
= 2( 1/1006 + 1/1008 + 1/1010 + ... + 1/2010) - (1- 1/2 + 1/3 - ... +1/1005)
= ( 1/503 + 1/504 + 1/505 +...+ 1/1005) - (1- 1/2 + 1/3 - ... +1/1005)
= (-1 + 1/2 - 1/3 + 1/4 - ... - 1/501 + 1/502) + 2(1/504 + 1/506+...+1/1004)
= (-1 + 1/2 - 1/3 + 1/4 - ... - 1/501 + 1/502) + (1/252 + 1/253 +... +1/502)
= (-1 + 1/2 - 1/3 + 1/4 - ... - 1/251) + 2(1/252 + 1/254 + 1/256 +...+1/502)
= (-1 + 1/2 - 1/3 + 1/4 - ... - 1/125) + 2(1/126 + 1/128+...+1/250)
= (-1 + 1/2 - 1/3 + 1/4 - ... - 1/61 + 1/62 ) + 2(1/64 +1/66 ...+1/124)
= (-1 + 1/2 - 1/3 + 1/4 - ... - 1/31) + 2(1/32+ 1/34 + ...+1/62)
= (-1 + 1/2 - 1/3 + 1/4 - ... - 1/15) + 2( 1/16 + 1/18 +...+1/30)
= (-1 + 1/2 - 1/3 + 1/4 - ... - 1/7) + 2(1/8 + 1/10 +... + 1/14)
=(-1 + 1/2 - 1/3 ) + 2(1/4+1/6)
= 0
所以兩式是相等的!
2010-12-12 15:59:45 補充：
我倒想知道 有沒有更快捷的方法去計算......
2010-12-14 5:28 pm
Sol
1－1/2+1/3－1/4+...+1/2009－1/2010
=(1+1/3+...+1/2009)－(1/2+1/4+...+1/2010)
=(1+1/3+...+1/2009)+(1/2+1/4+...+1/2010)－2*(1/2+1/4+...+1/2010)
=(1+1/2+1/3+1/4+...+1/2009+1/2010)－2*(1/2+1/4+...+1/2010)
=(1+1/2+1/3+1/4+...+1/2009+1/2010)－(1+1/2+1/3+1/4+...+1/1005)
=1/1006+1/1007+...+1/2009+1/2010
1－1/2+1/3－1/4+...+1/2009－1/2010
=(1+1/3+...+1/2009)－(1/2+1/4+...+1/2010)
=(1+1/3+...+1/2009)+(1/2+1/4+...+1/2010)－2*(1/2+1/4+...+1/2010)
=(1+1/2+1/3+1/4+...+1/2009+1/2010)－2*(1/2+1/4+...+1/2010)
=(1+1/2+1/3+1/4+...+1/2009+1/2010)－(1+1/2+1/3+1/4+...+1/1005)
=1/1006+1/1007+...+1/2009+1/2010
2010-12-14 3:26 am
= 2( 1/1006 + 1/1008 + 1/1010 + ... + 1/2010) - (1- 1/2 + 1/3 - ... +1/1005)
???
2010-12-13 19:26:27 補充：
1 - 1/2 + 1/3 - 1/4 + ... + 1/2009 - 1/2010 -(1)
1/1006 + 1/1007 + ... + 1/2009 + 1/2010 -(2)
(1) - (2) :
1 + 1/3 + 1/5 + ... + 1/1005 - 1/2 - 1/4 - 1/6 - ... - 1/1004 - 22/1006 - 2/1008 - ... - 2/2010
= 1 + 1/3 + 1/5 + ... + 1/1005 - 1/2 - 1/4 - 1/6 - ... - 1/1004 - 1/503 - 1/504 - ... - 1/1005
= 1 + 1/3 + 1/5 + ... + 1/501 - 1/2 - 1/4 - 1/6 - ... - 1/1004 - 1/504 - 1/506 - 1/508 - ... - 1/1002 - 1/1004
= 1 + 1/3 + 1/5 + ... + 1/501 - 1/2 - 1/4 - 1/6 - ... - 1/502 - 2/504 - 2/506 - 2/508 - ... - 2/1002 - 2/1004
= 1 + 1/3 + 1/5 + ... + 1/501 - 1/2 - 1/4 - 1/6 - ... - 1/502 - 1/252 - 1/253 - 1/254 - ... - 1/501 - 1/502
= 1 + 1/3 + 1/5 + ... + 1/251 - 1/2 - 1/4 - 1/6 - ... - 1/502 - 1/252 - 1/254 - 1/256 - ... - 1/500 - 1/502
= 1 + 1/3 + 1/5 + ... + 1/251 - 1/2 - 1/4 - 1/6 - ... - 1/250 - 1/126 - 1/127 - 1/128 - ... - 1/250 - 1/251
= 1 + 1/3 + 1/5 + ... + 1/125 - 1/2 - 1/4 - 1/6 - ... - 1/125 - 1/63 - 1/64 - ... - 1/124 - 1/125
= 1 + 1/3 + 1/5 + ... + 1/61 - 1/2 - 1/4 - 1/6 - ... - 1/125 - 1/64 - 1/66 - ... - 1/124
= 1 + 1/3 + 1/5 + ... + 1/31 - 1/2 - 1/4 - 1/6 - ... - 1/63 - 1/125 - 1/32 - 1/34 - ... - 1/60 - 1/62
= 1 + 1/3 + 1/5 + ... + 1/31 - 1/2 - 1/4 - 1/6 - ... - 1/30 - 1/63 - 1/125 - 1/16 - 1/17 - ... - 1/31
= 1 + 1/3 + 1/5 + ... + 1/15 - 1/2 - 1/4 - 1/6 - ... - 1/30 - 1/63 - 1/125 - 1/16 - 1/18 - ... - 1/28 - 1/30
= 1 + 1/3 + 1/5 + ... + 1/15 - 1/2 - 1/4 - 1/6 - ... - 1/14 - 1/63 - 1/125 - 1/8 - 1/9 - 1/10 - ... - 1/15
= 1 + 1/3 + 1/5 + 1/7 - 1/2 - 1/4 - 1/6 - ... - 1/14 - 1/63 - 1/125 - 1/8 - 1/10 - ... 1/14
= 1 + 1/3 + 1/5 + 1/7 - 1/2 - 1/4 - 1/6 - 1/63 - 1/125 - 1/4 - 1/5 - 1/6 - 1/7
= 1 + 1/3 - 1/2 - 1/2 - 1/3 - 1/63 - 1/125
= -1/63 - 1/125
(2) > (1)
???
2010-12-13 19:26:27 補充：
1 - 1/2 + 1/3 - 1/4 + ... + 1/2009 - 1/2010 -(1)
1/1006 + 1/1007 + ... + 1/2009 + 1/2010 -(2)
(1) - (2) :
1 + 1/3 + 1/5 + ... + 1/1005 - 1/2 - 1/4 - 1/6 - ... - 1/1004 - 22/1006 - 2/1008 - ... - 2/2010
= 1 + 1/3 + 1/5 + ... + 1/1005 - 1/2 - 1/4 - 1/6 - ... - 1/1004 - 1/503 - 1/504 - ... - 1/1005
= 1 + 1/3 + 1/5 + ... + 1/501 - 1/2 - 1/4 - 1/6 - ... - 1/1004 - 1/504 - 1/506 - 1/508 - ... - 1/1002 - 1/1004
= 1 + 1/3 + 1/5 + ... + 1/501 - 1/2 - 1/4 - 1/6 - ... - 1/502 - 2/504 - 2/506 - 2/508 - ... - 2/1002 - 2/1004
= 1 + 1/3 + 1/5 + ... + 1/501 - 1/2 - 1/4 - 1/6 - ... - 1/502 - 1/252 - 1/253 - 1/254 - ... - 1/501 - 1/502
= 1 + 1/3 + 1/5 + ... + 1/251 - 1/2 - 1/4 - 1/6 - ... - 1/502 - 1/252 - 1/254 - 1/256 - ... - 1/500 - 1/502
= 1 + 1/3 + 1/5 + ... + 1/251 - 1/2 - 1/4 - 1/6 - ... - 1/250 - 1/126 - 1/127 - 1/128 - ... - 1/250 - 1/251
= 1 + 1/3 + 1/5 + ... + 1/125 - 1/2 - 1/4 - 1/6 - ... - 1/125 - 1/63 - 1/64 - ... - 1/124 - 1/125
= 1 + 1/3 + 1/5 + ... + 1/61 - 1/2 - 1/4 - 1/6 - ... - 1/125 - 1/64 - 1/66 - ... - 1/124
= 1 + 1/3 + 1/5 + ... + 1/31 - 1/2 - 1/4 - 1/6 - ... - 1/63 - 1/125 - 1/32 - 1/34 - ... - 1/60 - 1/62
= 1 + 1/3 + 1/5 + ... + 1/31 - 1/2 - 1/4 - 1/6 - ... - 1/30 - 1/63 - 1/125 - 1/16 - 1/17 - ... - 1/31
= 1 + 1/3 + 1/5 + ... + 1/15 - 1/2 - 1/4 - 1/6 - ... - 1/30 - 1/63 - 1/125 - 1/16 - 1/18 - ... - 1/28 - 1/30
= 1 + 1/3 + 1/5 + ... + 1/15 - 1/2 - 1/4 - 1/6 - ... - 1/14 - 1/63 - 1/125 - 1/8 - 1/9 - 1/10 - ... - 1/15
= 1 + 1/3 + 1/5 + 1/7 - 1/2 - 1/4 - 1/6 - ... - 1/14 - 1/63 - 1/125 - 1/8 - 1/10 - ... 1/14
= 1 + 1/3 + 1/5 + 1/7 - 1/2 - 1/4 - 1/6 - 1/63 - 1/125 - 1/4 - 1/5 - 1/6 - 1/7
= 1 + 1/3 - 1/2 - 1/2 - 1/3 - 1/63 - 1/125
= -1/63 - 1/125
(2) > (1)
參考: Hope I Can Help You ^_^
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