We know about Euclid Number . I want to know the sum of reciprocals of 1st $n$ Euclid Number ? In that book I have been told to find the value of following series :
$ \dfrac{1}{e_1}+\dfrac{1}{e_2}+\cdots+\dfrac{1}{e_n} = ? $
In fact, I can not understand the following calculations: $ \dfrac{1}{e_1}+\dfrac{1}{e_2}+\cdots+\dfrac{1}{e_n} = 1 - \dfrac{1}{e_n(e_{n}-1)} = 1-\dfrac{1}{e_{n+1}-1}$
Can you please help me to find how this sum is done? This math is taken from "Concrete Math" Of Ronald L. Graham, Donald E. Knuth, and Oren Patashnik.
The definition in this book for Euclid numbers is non-standard: $e_1=2$ and $e_{n+1}=e_1\dots e_n +1$.