Question

Cho \(A=\dfrac{1}{2}+\dfrac{1}{301}+\dfrac{1}{302}+...+\dfrac{1}{399}+\dfrac{1}{400}\)

Chứng minh rằng \(A< 1\)

Answer 1

\(A=\dfrac{1}{2}+\dfrac{1}{301}+\dfrac{1}{302}+...+\dfrac{1}{400}\)

Do \(\dfrac{1}{301}< \dfrac{1}{300};\dfrac{1}{302}< \dfrac{1}{300};...;\dfrac{1}{400}< \dfrac{1}{300}\)

\(\Rightarrow A< \dfrac{1}{2}+\dfrac{1}{300}+\dfrac{1}{300}+...+\dfrac{1}{300}+\dfrac{1}{300}\) (100 số \(\dfrac{1}{300}\) )

\(\Rightarrow A< \dfrac{1}{2}+\dfrac{100}{300}\)

\(\Rightarrow A< \dfrac{1}{2}+\dfrac{1}{3}=\dfrac{5}{6}< 1\)

Vậy \(A< 1\)