1/(2根号1+1根号2)+1/(3根号2+2根号3)+.+1/(100根号99+99根号100)
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![1/(2根号1+1根号2)+1/(3根号2+2根号3)+.+1/(100根号99+99根号100)](/uploads/image/z/3987729-9-9.jpg?t=1%2F%282%E6%A0%B9%E5%8F%B71%2B1%E6%A0%B9%E5%8F%B72%29%2B1%2F%283%E6%A0%B9%E5%8F%B72%2B2%E6%A0%B9%E5%8F%B73%29%2B.%2B1%2F%28100%E6%A0%B9%E5%8F%B799%2B99%E6%A0%B9%E5%8F%B7100%29)
1/(2根号1+1根号2)+1/(3根号2+2根号3)+.+1/(100根号99+99根号100)
1/(2根号1+1根号2)+1/(3根号2+2根号3)+.+1/(100根号99+99根号100)
1/(2根号1+1根号2)+1/(3根号2+2根号3)+.+1/(100根号99+99根号100)
1/(2根号1+1根号2)
=(2根号1-1根号2)/(4-2)
=根号1-(根号2)/2
1/(3根号2+2根号3)
=(3根号2-2根号3)/(18-12)
=(根号2)/2-(根号3)/3
1/(2根号1+1根号2)+1/(3根号2+2根号3)+.+1/(100根号99+99根号100)
=根号1-(根号)/2+(根号)/2-(根号3)/3+(根号)/3-...+(根号99)/99-(根号100)/100
=根号1-(根号100)/100
=1-1/10
=9/10
1/(2+根号2)=(2-根号2)/(2+根号2)*(2-根号2)
=(2-根号2)/2
=1-(根号2)/2
1/(3根号2+2根号3)=(3根号2-2根号3)/(3根号2-2根号3)*(3根号2+2根号3) =(3根号2-2根号3)/6
=(根号2)/2 -(根号3)/3
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1/(2+根号2)=(2-根号2)/(2+根号2)*(2-根号2)
=(2-根号2)/2
=1-(根号2)/2
1/(3根号2+2根号3)=(3根号2-2根号3)/(3根号2-2根号3)*(3根号2+2根号3) =(3根号2-2根号3)/6
=(根号2)/2 -(根号3)/3
同理,1/(100根号99+99根号100)=(根号99)/99 -(根号100)/100
所以, 1/(2根号1+1根号2)+1/(3根号2+2根号3)+.......+1/(100根号99+99根号100)
=[1-(根号2)/2]+[(根号2)/2-(根号3)/3]+......+[(根号99)/99 -(根号100)/100]
=1-(根号100)/100
=1-1/10=0.9
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