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a) A= 1/2010+1+2/2009+1+3/2008+1+...+2009/2+1+1
= 2011/2010+20011/2009+2011/2008+...+2011/2+2011/2011
= 2011(1/2+1/3+1/4+...+1/2011)
Ta có: B= 1/2+1/3+1/4+...+1/2011
suy ra A/B= 2011
\(b)\) Ta có công thức :
\(\frac{a}{b}< \frac{a+c}{b+c}\)\(\left(a,b,c\inℕ^∗\right)\)
Áp dụng vào ta có :
\(\frac{2009^{2010}-2}{2009^{2011}-2}< \frac{2009^{2010}-2+2011}{2009^{2011}-2+2011}=\frac{2009^{2010}+2009}{2009^{2011}+2009}=\frac{2009\left(2009^{2009}+1\right)}{2009\left(2009^{2010}+1\right)}=\frac{2009^{2009}+1}{2009^{2010}+1}\)
Vậy \(\frac{2009^{2009}+1}{2009^{2010}+1}>\frac{2009^{1010}-2}{2009^{2011}-2}\)
Chúc bạn học tốt ~
Àk mình còn thiếu một điều kiện nữa xin lỗi nhé :
Ta có công thức :
\(\frac{a}{b}< \frac{a+c}{b+c}\)\(\left(\frac{a}{b}< 1;a,b,c\inℕ^∗\right)\)
Bạn thêm vào nhé
\(B=\frac{2009^{2010}-2}{2009^{2011}-2}< 1\)
\(\Rightarrow B=\frac{2009^{2010}-2}{2009^{2011}-2}< \frac{2009^{2010}-2+2011}{2009^{2011}-2+2011}=\frac{2009^{2010}+2009}{2009^{2011}+2009}\)\(=\frac{2009.\left(2009^{2009}+1\right)}{2009.\left(2009^{2010}+1\right)}=\frac{2009^{2009}+1}{2009^{2010}+1}\)
Suy ra : \(\frac{2009^{2010}-2}{2009^{2011}-2}< \frac{2009^{2009}+1}{2009^{2010}+1}\) hay \(B< A\)
Vậy \(A>B\)
Ta có:
\(B=\dfrac{2009^{2010}-2}{2009^{2011}-2}\)
\(B< \dfrac{2009^{2010}-2+2011}{2009^{2011}-2+2011}\)
\(B< \dfrac{2009^{2010}+2009}{2009^{2011}+2009}\)
\(B< \dfrac{2009\left(2009^{2009}+1\right)}{2009\left(2009^{2010}+1\right)}\)
\(B< \dfrac{2009^{2009}+1}{2009^{2010}+1}\)
Mà \(A=\dfrac{2009^{2009}+1}{2009^{2010}+1}\)
\(\Rightarrow B< A\)
\(A=\frac{2010}{1}+\frac{2009}{2}+...+\frac{2}{2009}+\frac{1}{2010}\)
\(A=1+\left(\frac{2009}{2}+1\right)+...+\left(\frac{2}{2009}+1\right)+\left(\frac{1}{2010}+1\right)\)
\(A=\frac{2011}{2011}+\frac{2011}{2}+...+\frac{2011}{2009}+\frac{2011}{2010}\)
\(A=\frac{2011}{2}+...+\frac{2011}{9}+\frac{2011}{10}+\frac{2011}{11}\)
\(A=2011.\left(\frac{1}{2}+...+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}\right)\)
\(A=2011.B\)
Nên : \(\frac{A}{B}=\frac{2011.B}{B}=2011\)
Vậy \(\frac{A}{B}=2011\)
Tham khảo nha !!! Chúc bạn học tốt !!!