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.

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