\(P< \frac{2}{3}.\frac{4}{5}.\frac{6}{7}...\frac{400}{401}\)

\(P^2< \frac{1.2.3...400}{2.3.4...401}=\frac{1}{401}< \frac{1}{400}\)

\(\Rightarrow P< \frac{1}{20}\)

Đặt \(Q=\frac{2}{3}.\frac{4}{6}.\frac{6}{7}....\frac{400}{401}\)

Áp dụng tính chất \(\frac{a}{b}< \frac{a+m}{b+m}\left(a,b,m\inℕ^∗\right)\)ta có :

\(\frac{1}{2}< \frac{1+1}{2+1}=\frac{2}{3}\)

\(\frac{2}{3}< \frac{2+1}{3+1}=\frac{3}{4}\)

...

\(\frac{399}{400}< \frac{399+1}{400+1}=\frac{400}{401}\)

\(\Rightarrow\frac{1}{2}.\frac{3}{4}.\frac{5}{6}....\frac{399}{400}< \frac{2}{3}.\frac{4}{5}.\frac{6}{7}....\frac{400}{401}\)

Hay \(P< Q\)

\(\Rightarrow P^2< P.Q\)

    \(P^2< \frac{1}{2}.\frac{3}{4}.\frac{5}{6}...\frac{399}{400}.\frac{2}{3}.\frac{4}{5}.\frac{6}{7}....\frac{400}{401}\)

   \(P^2< \frac{1.2.3.4.....400}{2.3.4.5.....401}\)

   \(P^2< \frac{1}{401}< \frac{1}{400}< \left(\frac{1}{20}\right)^2\)

Vì \(P\)và  \(\frac{1}{2}\)có cùng dấu 

\(\Rightarrow P< \frac{1}{2}\)

                                                           Hk tốt 

p=1/2.3/4.5/6......399/400

=>p<1/2.2/4.4/6....398/400

p<1.2.4.....398/2.4.6....400

rut gon dc p<1/400<1/20

vay p < 1/20

Đặt Q =\(\frac{2}{3}.\frac{4}{5}.\frac{6}{7}.\frac{8}{9}.....\frac{400}{401}\)

Dễ thấy: P < Q

Mặt khác:

P.Q = \(\frac{1}{2}.\frac{2}{3}.\frac{3}{4}.\frac{4}{5}....\frac{399}{400}.\frac{400}{401}=\frac{1.2.3....399.400}{2.3.4...400.401}\)

=\(\frac{1}{401}< \frac{1}{400}=\left(\frac{1}{20}\right)^2\)

Mà \(P^2< P.Q< \left(\frac{1}{20}\right)^2\Leftrightarrow P< \frac{1}{20}\)

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Các PS 1/201, 1/202,..1/399 đều lớn hơn

Đặt \(Q=\frac{2}{3}.\frac{4}{5}.\frac{6}{7}.....\frac{400}{401}\)

Áp dụng tính chất \(\frac{a}{b}< \frac{a+m}{b+m}\left(a,b,m\inℕ^∗\right)\)ta có

\(\frac{1}{2}< \frac{1+1}{2+1}=\frac{2}{3}\)

\(\frac{2}{3}< \frac{2+1}{3+1}=\frac{3}{4}\)

...

\(\frac{399}{400}< \frac{399+1}{400+1}=\frac{400}{401}\)

\(\Rightarrow\frac{1}{2}.\frac{3}{4}.\frac{5}{6}.....\frac{399}{400}< \frac{2}{3}.\frac{4}{5}.\frac{6}{7}.....\frac{400}{401}\)

hay P < Q

=> \(P^2< P.Q\)

      \(P^2< \frac{1}{2}.\frac{3}{4}.\frac{5}{6}.....\frac{399}{400}.\frac{2}{3}.\frac{4}{5}.\frac{6}{7}.....\frac{400}{401}\)

       \(P^2< \frac{1.2.3.4.....400}{2.3.4.5.....401}\)

        \(P^2< \frac{1}{401}< \frac{1}{400}< \left(\frac{1}{20}\right)^2\)

Vì P và 1/20 có cùng dấu

\(\Rightarrow P< \frac{1}{20}\)