有更快的办法计算吗？

lavendar_o5

1 + 1/(1 + 2) + 1/(1 + 2 + 3) + 1/(1 + 2 + 3 + 4) + ... + 1/(1+2+3+4+5+6+7+8+9+10)

mathlim

1 + 2 + 3 + ... ... + n = n(n+1)/2

1/(1 + 2 + 3 + ... ... + n) = 2/n(n+1)

而 2/n(n+1) = 2/n - 2/(n+1)

∴ 1 + 1/(1+2) + 1/(1+2+3) + ... ... + 1/(1+2+3+4+5+6+7+8+9+10)
= [2/1 - 2/2] + [2/2 - 2/3] + [2/3 - 2/4] + ... ... + [2/10 - 2/11]
= 2 - 2/11
= 20/11

通项公式：
  1 + 1/(1+2) + 1/(1+2+3) + ... ... + 1/(1+2+3+... ...+n)
= [2/1 - 2/2] + [2/2 - 2/3] + [2/3 - 2/4] + ... ... + [2/n - 2/(n+1)]
= 2 - 2/(n+1)
= 2n/(n+1)

当 n = 10，
  1 + 1/(1+2) + 1/(1+2+3) + ... ... + 1/(1+2+3+4+5+6+7+8+9+10)
= 2×10/(10+1)
= 20/11

mathlim

再教你一个方法，就是不要贪心，一步一步来。

1 = 2/2
1 + 1/3 = 4/3
1 + 1/3 + 1/6 = 4/3 + 1/6 = 9/6 = 3/2 = 6/4
1 + 1/3 + 1/6 + 1/10 = 3/2 + 1/10 = 16/10 = 8/5
1 + 1/3 + 1/6 + 1/10 + 1/15 = 8/5 + 1/15 = 25/15 = 5/3 = 10/6
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归纳出：
1 + 1/3 + 1/6 + ... ... + 1/(1+2+3+... ...+n) = 2n/(n+1)
∴ 1 + 1/3 + 1/6 + ... ... + 1/(1+2+3+... ...+10) = 20/11