Question

108+(-178)+128+(-13.3)

简算：1/(2²-1)+1/（3²-1）+1/(4²-1）+...+1/(20²-1）=
1 1 1 1 1 1
--- + ------- +--------- +--------+.+- ---------- +---------- =
2²-1 3²-1 4²-1 5²-1 19²-1 20²-1

Answer 1

1/(2²-1)+1/(3²-1)+1/(4²-1)+...+1/(20²-1)
= 1/[(2-1)(2+1)]+1/[(3-1)(3+1)]+1/[(4-1)(4+1)]+...+1/[(20-1)(20+1)]
= 1/(1×3)+1/(2×4)+1/(3×5)+...+1/(19×21)
= (1/2)×[(1-1/3)+(1/2-1/4)+(1/3-1/5)+...+(1/19-1/21)]
= (1/2)×[1+1/2-1/20-1/21]
= (1/2)×[(420+210-21-20)/420]
= 589/840