我有一个计算器,“0”键已坏。我使用它作自然数倒数的加法:1/1+1/2+1/3.....,当然凡包括数字0的数n,其倒数1/n都不能加进去,如1/10,1/101等等。证明:不管我连续加多长时间(哪怕加到老死),得到的和数不会达到100 。(自然数倒数之和是发散的,地球人都知道。) |
你的意思是 这个和数是收敛的,而且一定小于 100,对吗? |
Consider ln(x) = 100, x = 2.688 * 10^43. Consider your calculator skill is excellent, you could key in 10 numbers per second. You need 8.5 * 10^34 (estimate) years to make the sum go over 100. If you could live long enough, try it, you will get it. |
For numbers between 10^n and 10^(n+1), 9^(n+1) of them have no 0 digits. Hence the contribution of all these numbers toward the sum is bounded by: 9 (9/10)^n, and 9(9/10)^0+9(9/10)^1+..... = =90 |
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