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Recently Referred Formulae

\frac{2.0+1.9}{(1.8-1.7)\times(1.6-1.5)\times(1.4-1.3)\times(1.2-1.1)}=39000
\ln x\le x-1
\frac{1}{\ln r}\left(\sum_{i=1}^MP_i-\sum_{i=1}^Mq_i\right)=\frac{1}{\ln r}\left(1-\sum_{i=1}^Mq_i\right)\ge1
\frac{1}{\ln r}\sum_{i=1}^MP_i\left(1-\frac{q_i}{P_q}\right)=\frac{1}{\ln r}\left(\sum_{i=1}^MP_i-\sum_{i=1}^Mq_i\right)
-\sum_{i=1}^M\left(P_i\log\frac{q_i}{P_i}\right)\ge\frac{1}{\ln r}\sum_{i=1}^MP_i\left(1-\frac{q_i}{P_i}\right)\\