ワイエルシュトラスの因数分解定理
= − z n + 1 n + 1 − z n + 2 n + 2 − z n + 3 n + 3 − ⋯ = − ∑ k = n + 1 ∞ z n + 1 n + 1 = − z n + 1 n + 1 ∑ k = 0 ∞ n + 1 n + 1 + k z k {\displaystyle =-{\tfrac {z^{n+1}}{n+1}}-{\tfrac {z^{n+2}}{n+2}}-{\tfrac {z^{n+3}}{n+3}}-\cdots =-\sum _{k=n+1}^{\infty }{\tfrac {z^{n+1}}{n+1}}=-{\tfrac {z^{n+1}}{n+1}}\sum _{k=0}^{\infty }{\tfrac {n+1}{n+1+k}}z^{k}}
従って、
| log  E n ( z ) | < | z | n + 1 n + 1 ∑ k = 0 ∞ n + 1 n + 1 + k | z | k ≤ | z | n + 1 n + 1 ∑ k = 0 ∞ | z | k = | z | n + 1 n + 1 1 1 − | z | {\displaystyle |\log E_{n}(z)|<{\tfrac {|z|^{n+1}}{n+1}}\sum _{k=0}^{\infty }{\tfrac {n+1}{n+1+k}}|z|^{k}\leq {\tfrac {|z|^{n+1}}{n+1}}\sum _{k=0}^{\infty }|z|^{k}={\tfrac {|z|^{n+1}}{n+1}}{\tfrac {1}{1-|z|}}}