Question

[Hoàng Việt Anh]

cho A=1+1\2+1\3+...+1\4026,B=1+1\3+1\5+...+1\4025.So sánh A\B và 1\2013\2014

Answer 1

[T.Anh 2K7(siêu quậy)(тoáɴ๖ۣۜнọc)]

\(\Rightarrow A-B=\frac{1}{2}+\frac{1}{4}+...+\frac{1}{4026}\)

\(B>1+\frac{1}{4}+\frac{1}{6}+...+\frac{1}{4026}=\frac{1}{2}+\left(\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+...+\frac{1}{4026}\right)=\frac{1}{2}+\left(A-B\right)\)

\(\Rightarrow B>\frac{1}{2}+\left(A-B\right)\left(1\right)\)

\(A-B=\frac{1}{2}+\frac{1}{4}+...+\frac{1}{4026}< \frac{1}{2}+\frac{1}{2}+...+\frac{1}{2}=\frac{2013}{2}\)

\(\Rightarrow A-B< \frac{2013}{2}\Rightarrow\frac{A-B}{2013}< \frac{1}{2}\left(2\right)\)

Cộng (1) với (2)

\(\Rightarrow\frac{A-B}{2013}+\frac{1}{2}+\left(A-B\right)< \frac{1}{2}+B\Rightarrow\frac{A-B}{2013}+\left(A-B\right)< B\Rightarrow\frac{2014\left(A-B\right)}{2013}< B\Rightarrow\frac{A-B}{B}< \frac{2013}{2014}\)

\(\Rightarrow\frac{A-B}{B}+1< \frac{2013}{2014}+1\Rightarrow\frac{A}{B}< 1\frac{2013}{2014}\left(đpcm\right)\)