2000/2001 < 2001/2002

ta có\(\dfrac{2001}{2002}< 1\)và\(\dfrac{2005}{2003}>1\)

\(\Rightarrow\dfrac{2001}{2002}< \dfrac{2005}{2003}\)

Ta có: \(\dfrac{2001}{2002}< 1\)

mà \(1< \dfrac{2005}{2003}\)

nên \(\dfrac{2001}{2002}< \dfrac{2005}{2003}\)

ta có \(\frac{2000+2002}{2001+2003}\)= \(\frac{2000}{2001+2003}\)+ \(\frac{2002}{2001+2003}\)=\(\frac{2000}{4004}\)+\(\frac{2002}{4004}\)

ta có \(\frac{2000}{2001}\)>\(\frac{2000}{4004}\) và \(\frac{2002}{2003}\)> \(\frac{2002}{4004}\)

 nên \(\frac{2000}{2001}\)+\(\frac{2002}{2003}\)>\(\frac{2000}{4004}\)+\(\frac{2002}{4004}\)

vậy \(\frac{2000}{2001}\)+\(\frac{2002}{2003}\)>\(\frac{2000+2002}{2001+2003}\)

\(\frac{2000+2002}{2001+2003}=\frac{2000}{2001+2003}+\frac{2002}{2001+2003}< \frac{2000}{2001}+\frac{2002}{2003}\)

Tham khảo : 

Sứa , san hô , hải quỳ , thủy tức , sứa tu dài ,...

\(\dfrac{2001+2002}{2002+2003}< \dfrac{2001}{2002}+\dfrac{2002}{2003}\)

B = \(\frac{2001}{2002}+\frac{2002}{2003}\)

có: \(\frac{2000}{2001}>\frac{2000}{2001}+2002\)

\(\frac{2001}{2002}>\frac{2001}{2001}+2002\)

Vậy A>B

So sánh:

Ta có:\(\hept{\begin{cases}\frac{2002}{2003}< \frac{2003}{2003}=1\\\frac{14}{13}>\frac{13}{13}=1\end{cases}}\)

\(\Rightarrow\frac{14}{13}>\frac{2002}{2003}\)