# Ngô Quốc Anh

## October 16, 2007

### 1 = …?

Filed under: Các Bài Tập Nhỏ — Ngô Quốc Anh @ 23:12

$S=1$
$=1\left(\boxed{\frac{1}{2}}+\frac{1}{2}\right)$
$=1\left(\frac{1}{2}+\frac{1}{2}\left(\boxed{\frac{1}{3}}+\frac{2}{3}\right)\right)$
$=1\left(\frac{1}{2}+\frac{1}{2}\left(\frac{1}{3}+\frac{2}{3}\left(\boxed{\frac{1}{5}}+\frac{4}{5}\right)\right)\right)$
$=1\left(\frac{1}{2}+\frac{1}{2}\left(\frac{1}{3}+\frac{2}{3}\left(\frac{1}{5}+\frac{4}{5}\left(\boxed{\frac{1}{7}}+\frac{6}{7}\right)\right)\right)\right)$
$=1\left(\frac{1}{2}+\frac{1}{2}\left(\frac{1}{3}+\frac{2}{3}\left(\frac{1}{5}+\frac{4}{5}\left(\frac{1}{7}+\frac{6}{7}\left(\boxed{\frac{1}{11}}+\frac{10}{11}\right)\right)\right)\right)\right)$
$=1\left(\frac{1}{2}+\frac{1}{2}\left(\frac{1}{3}+\frac{2}{3}\left(\frac{1}{5}+\frac{4}{5}\left(\frac{1}{7}+\frac{6}{7}\left(\frac{1}{11}+\frac{10}{11}\left(\boxed{\frac{1}{13}}+\frac{12}{13}\right)\right)\right)\right)\right)\right)\right)$
$=1\left(\frac{1}{2}+\frac{1}{2}\left(\frac{1}{3}+\frac{2}{3}\left(\frac{1}{5}+\frac{4}{5}\left(\frac{1}{7}+\frac{6}{7}\left(\frac{1}{11}+\frac{10}{11}\left(\frac{1}{13}+\frac{12}{13}\left(\boxed{\frac{1}{17}}+\frac{16}{17}\right)\right)\right)\right)\right)\right)\right)\right)$
$\ldots\;\;\;\ldots$

$S=\sum_{n=1}^\infty\;\left\{\frac{1}{p_n-1}\prod_{k=1}^n\;\left(1-\frac{1}{p_k}\right)\right\}$

the thing is anyone can basically start from $1$ and go on to concoct any type of series and then ask for it’s sum.