1\3+1\15+1\35+1\63+1\99

1\3+1\15+1\35+1\63+1\99
1\3+1\15+1\35+1\63+1\99

1\3+1\15+1\35+1\63+1\99
1/3+1/15+1/35+1/63+1/99
=[（1-1/3）+（1/3-1/5）+（1/5-1/7）+（1/7-1/9）+（1/9-1/11）]÷(3-1)
=[1-1/3+1/3-1/5+1/5-1/7+1/7-1/9+1/9-1/11]÷2
=(1-1/11)÷2
=5/11
裂项分拆:
1/(a×b)=(1/a-1/b)÷(b-a)

原式=(1-1/3)×1/2+(1/3-1/5) ×1/2+(1/5-1/7) ×1/2+(1/7-1/9) ×1/2+(1/9-1/11) ×1/2
=(1-1/3+1/3-1/5+1/5-1/7+1/7-1/9+1/9-1/11) ×1/2
=10/11×1/2=5/11

1\3+1\15+1\35+1\63+1\99
=1\(1*3)+1\(3*5)+1\(5*7)+1\(7*9)+1\(9*11)
=(1-1\3+1\3-1\5+1\5-1\7+1\7-1\9+1\9-1\11)\2
=5\11