Lehmer的有趣系列
Lehmer's Interesting Series
摘要：
系列$$S_k(z)=\sum_{m=1}^{\infty}\frac{m^kz^m}{(\{array}{c} 2m m \{array})}$$评估非递归封闭和解析连续超出其域的收敛$0\le |z|<4$为$k=0,1,2,\...$。由此，我们为Lehmer的观察提供了坚实的基础，即$π$是从$S_k(2)$的极限行为中作为$k\rightarrow\infty$出现的。
英文摘要：
The series $$S_k(z)=\sum_{m=1}^{\infty}\frac{m^kz^m}{(\{array}{c} 2m m \{array})}$$ is evaluated in non-recursive closed and analytically continued beyond its domain of convergence $0\le |z|<4$ for $k=0,1,2,\...$. From this we provide a firm basis for Lehmer's observation that $π$ emerges from the limiting behavior of $S_k(2)$ as $k\rightarrow\infty$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
