求和Kempner与Irwin的奇特级数
Summing the curious series of Kempner and Irwin
AI总结：
本文研究了Kempner和Irwin提出的分母不含或含有限个特定数字的收敛级数，提出了一种高精度计算Irwin级数和的方法，并讨论了尚未开发求和算法的相关级数。
摘要：
1914年，Kempner证明了级数1/1 + 1/2 + ... + 1/8 + 1/10 + 1/11 + ... + 1/18 + 1/20 + 1/21 + ...（其分母为不含数字9的正整数）收敛，且其和小于90。实际和约为22.92068。1916年，Irwin证明了（除其他结论外）1/n的和（其中n最多包含有限个9）也是一个收敛级数。我们展示了如何高精度地计算Irwin级数的和。例如，级数1/9 + 1/19 + 1/29 + 1/39 + 1/49 + ...（其分母恰好包含一个9）的和约为23.04428 70807 47848 31968。另一个例子：1/n的和（其中n恰好包含100个0）约为10 ln(10) + 1.00745 x 10^-197 ~ 23.02585；注意，该级数的第一项，也是最大项，是极小的1/googol。最后，我们讨论了一类尚未开发求和算法的相关级数。
英文摘要：
In 1914, Kempner proved that the series 1/1 + 1/2 + ... + 1/8 + 1/10 + 1/11 + ... + 1/18 + 1/20 + 1/21 + ... where the denominators are the positive integers that do not contain the digit 9, converges to a sum less than 90. The actual sum is about 22.92068. In 1916, Irwin proved, among other things, that the sum of 1/n where n has at most a finite number of 9's is also a convergent series. We show how to compute sums of Irwins' series to high precision. For example, the sum of the series 1/9 + 1/19 + 1/29 + 1/39 + 1/49 + ... where the denominators have exactly one 9, is about 23.04428 70807 47848 31968. Another example: the sum of 1/n where n has exactly 100 zeros is about 10 ln(10) + 1.00745 x 10^-197 ~ 23.02585; note that the first, and largest, term in this series is the tiny 1/googol. Finally, we discuss a class of related series whose summation algorithm has not yet been developed.
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