e 극한 곱 로그 합

$\displaystyle \begin{aligned} P _{n} & = \lim\limits _{x \rightarrow \infty} {} \left( 1 + \frac{1}{x} \right) ^{x} \left( 1 + \frac{2}{x} \right) ^{x} \left( 1 + \frac{3}{x} \right) ^{x} \cdots \left( 1 + \frac{n}{x} \right) ^{x} \end{aligned}$ $\displaystyle = \lim\limits _{x \rightarrow \infty} {} \left( 1 + \frac{1}{x} \right) ^{x} \left\{ \left( 1 + \frac{2}{x} \right) ^{\frac{x}{2}} \right\} ^{2} \left\{ \left( 1 + \frac{3}{x} \right) ^{\frac{x}{3}} \right\} ^{3} \cdots \left\{ \left( 1 + \frac{n}{x} \right) ^{\frac{x}{n}} \right\} ^{n}$ $\displaystyle = e \cdot e ^{2} \cdot e ^{3} \cdot \cdots \cdot e ^{n} = e ^{1 + 2 + 3 + \cdots + n} = e ^{\frac{n ( n + 1 )}{2}}$ $\displaystyle \ln P _{n} = \frac{n ( n + 1 )}{2}$ $\displaystyle \therefore \sum\limits _{{n} = 1} ^{20} \frac{1}{\ln Pn} = \sum\limits _{{n} = 1} ^{20} \frac{2}{n ( n + 1 )} = 2 \left( 1 - \frac{1}{21} \right) = \frac{40}{21}$