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第9期问題解答(解答由提出人給出) 493.从調和級数 1/1+1/2+1/3+…+1/n+…里划去所有分母中含有数字9的那些項(例如1/9,1/199,…,1/1093等)之后,所組成的部分新級数是收斂到一个不超过80的数。 証.規定部分級数中,分母中各数仅由0,1,2,…,8这9个数字所組成。所以介于和之間所有各数之总数就相当于将9个元素每次取m个的所有可能的重复的选排列数即9~m那么多种可能。这样,包含在10~(m-1)-1与10~(m-1)之間而不含有数字9的非負数为9~m-9~(m-1)个。于是有那个新級数的全体为
The 9th question answer (the answer is given by the proposer) 493. From the harmonic progression 1/1+1/2+1/3+... + 1/n+..., remove all the items with the number 9 in the denominator (eg, After 1/9, 1/199, ..., 1/1093, etc.), the new part of the composition is a number that does not exceed 80. In the prescribed part of the series, the numbers in the denominator consist of only 9 digits 0, 1, 2, ..., and 8. Therefore, the sum of all the numbers between and is equivalent to the number of possible repeatable selections of 9 elements at a time, ie, 9 to m. In this way, the non-negative number contained between 10~(m-1)-1 and 10~(m-1) without the number 9 is 9~m-9~(m-1). So the whole new series is