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.

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.
由1/(2√1+1√2)+1/(3√2+2√3)+1/(4√3+3√4)+...+1/(100√99+99√100)
=(2√1-1√2)/(2√1+1√2)(2√1-1√2)+(3√2-2√3)/(3√2+2√3)(3√2-2√3)
+(4√3-3√4)/(4√3+3√4)(4√3-3√4)+...+(100√99-99√100)/(100√99+99√100)(100√99-99√100)
=(2-√2)/1×2+(3√2-2√3)/2×3+(4√3-3√4)/3×4+...+(100√99-99√100)/99×100
=(1-√2/2)+(√2/2-√3/3)+(√3/3-√4/4)+...+(√99/99-√100/100)
=1-√100/100
=1-1/10
=9/10.