1/(2×4)+1/(4×6)+1/(6×8)+1/(8×10)+···+1/(2006×2008)
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![1/(2×4)+1/(4×6)+1/(6×8)+1/(8×10)+···+1/(2006×2008)](/uploads/image/z/5509632-48-2.jpg?t=1%2F%EF%BC%882%C3%974%EF%BC%89%2B1%2F%EF%BC%884%C3%976%EF%BC%89%2B1%2F%EF%BC%886%C3%978%EF%BC%89%2B1%2F%EF%BC%888%C3%9710%EF%BC%89%2B%C2%B7%C2%B7%C2%B7%2B1%2F%EF%BC%882006%C3%972008%EF%BC%89)
1/(2×4)+1/(4×6)+1/(6×8)+1/(8×10)+···+1/(2006×2008)
1/(2×4)+1/(4×6)+1/(6×8)+1/(8×10)+···+1/(2006×2008)
1/(2×4)+1/(4×6)+1/(6×8)+1/(8×10)+···+1/(2006×2008)
1/(2×4)+1/(4×6)+1/(6×8)+1/(8×10)+···+1/(2006×2008)=
1/2 X [1/2-1/4 + 1/4-1/6 + 1/6-1/8 +.+1/2006-1/2008] =
1/2 X [1/2 - 1/2008] =
1/2 X (1004-1)/2008 = 1003/4016
1003/2008
这个是可以化简成
(1/2-1/4)/2+(1/4-1/6)/2....(1/2006-1/2008)/2这种的
所以就等于(1/2-1/4+1/4-1/6+1/6-1/8+.....-1/2008)/2
最后就是(1/2-1/2008)/2 =1003/4016
1/(2×4)=(1/2-1/4)*1/2
1/(4×6)=(1/4-1/6)*1/2
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1/(2006×2008)=(1/2006-1/2008)*1/2
所以原式=(1/2-1/2008)*1/2=结果
1/(2×4)+1/(4×6)+1/(6×8)+1/(8×10)+···+1/(2006×2008)=1/2(1/2-1/4)+1/2(1/4-1/6)+1/2(1/6-1/8)+...+1/2(1/2006-1/2008)
展开等式,中间的消除,即剩首项和尾项:
=1/4-1/4016=1003/4016
1/(2×4)+1/(4×6)+1/(6×8)+1/(8×10)+···+1/(2006×2008)
=(1/2)(1/2-1/4+1/4-1/6+...+1/2006-1/2008)
=(1/2)(1/2-1/2008)
=1003/4016