\(S=\frac{1}{5}+\left(\frac{1}{13}+\frac{1}{14}+\frac{1}{15}\right)+\left(\frac{1}{61}+\frac{1}{62}+\frac{1}{63}\right)\)

\(\Rightarrow S< \frac{1}{5}+\frac{1}{12}.3+\frac{1}{60}.3\)

\(\Rightarrow S< \frac{1}{5}+\frac{1}{4}+\frac{1}{20}\)

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

Đặt :

\(A=\)\(\dfrac{1}{5}+\dfrac{1}{13}+\dfrac{1}{14}+\dfrac{1}{15}+\dfrac{1}{61}+\dfrac{1}{62}+\dfrac{1}{63}\)

\(A=\dfrac{1}{5}+\left(\dfrac{1}{13}+\dfrac{1}{14}+\dfrac{1}{15}\right)+\left(\dfrac{1}{61}+\dfrac{1}{62}+\dfrac{1}{63}\right)\)

Ta thấy :

\(\dfrac{1}{13}+\dfrac{1}{14}+\dfrac{1}{15}< \dfrac{1}{12}+\dfrac{1}{12}+\dfrac{1}{12}\)

\(\dfrac{1}{61}+\dfrac{1}{62}+\dfrac{1}{63}< \dfrac{1}{60}+\dfrac{1}{61}+\dfrac{1}{62}\)

\(\Rightarrow A< \dfrac{1}{5}+\left(\dfrac{1}{12}+\dfrac{1}{12}+\dfrac{1}{12}\right)+\left(\dfrac{1}{60}+\dfrac{1}{60}+\dfrac{1}{60}\right)\)

\(\Rightarrow A< \dfrac{1}{5}+\dfrac{1}{12}.3+\dfrac{1}{60}.3\)

\(\Rightarrow A< \dfrac{1}{5}+\dfrac{1}{4}+\dfrac{1}{20}\)

\(\Rightarrow A< \dfrac{10}{20}=\dfrac{1}{2}\)

\(\Rightarrow A< \dfrac{1}{2}\rightarrowđpcm\)

A = 1/5 + 1/13 + 1/14 + 1/15 + 1/60 + 1/61 + 1/62 + 1/63

Ta có : A = 1/5 + 1/13 + 1/14 + 1/15 + 1/60 + 1/61 + 1/62 + 1/63 < 1/5 + 1/12 + 1/12 + 1/12 + 1/60 + 1/60 + 1/60 

               = A < 1/5 + 1/4 + 1/20 

               = A < 1/2

Vậy A < 1/12

S=1/5+(1/13+1/14+1/15)+(1/61+1/62+1/63)

suy ra S<1/5+1/12.3+1/60.3

S<1/5+1/4+1/20

S<1/2 

S<1/2

S=\(\frac{1}{5}\)+(\(\frac{1}{13}+\frac{1}{14}+\frac{1}{15}\)) + (\(\frac{1}{61}+\frac{1}{62}+\frac{1}{63}\))

=> S< \(\frac{1}{5}+\frac{1}{12}.3+\frac{1}{60}.3\)

S<\(\frac{1}{5}+\frac{1}{4}+\frac{1}{20}\)

=> S< \(\frac{1}{2}\)

Vậy S<\(\frac{1}{2}\)

 S=1/5+ 1/13+1/14+1/15+1/61+1/62+1/63< 1/2

S = 1/5 + ( 1/13 + 1/14 + 1/15 ) + ( 1/ 61 + 1/ 62 + 1/ 63 )

=> S < 1/5 + 1/12 . 3 + 1/ 60 . 3

=> S < 1/5 + 1/4 + 1/20

=> S < 1/2

Vậy S < 1/2