HMMT 二月 2008 · 代数 · 第 7 题

英文解析

7. [ 5 ] Compute .
   n + kn =12
   k =1
   Answer: We change the order of summation:41
   ∞ n − 1 ∞ ∞ ∞9
   ∑ ∑ ∑ ∑ ∑
   k k 1 k 4 = = = .
   n + k k n k
   2 2 2 4 9
   n =1
   k =1 k =1 n = k +1 k =1
   (The last two steps involve the summation of an infinite geometric series, and what is sometimes calledan infinite arithmetico-geometric series. These summations are quite standard, and thus we omit thedetails here.)
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