Question

So sánh A = 1/31 + 1/32 + 1/33 + ... + 1/89 + 1/90 với B = 5/6

Answer 1

Ta có: A= \(\dfrac{1}{31}+\dfrac{1}{32}+...+\dfrac{1}{90}\)

\(A=\left(\dfrac{1}{31}+\dfrac{1}{32}+...+\dfrac{1}{60}\right)+\left(\dfrac{1}{61}+\dfrac{1}{62}+...+\dfrac{1}{90}\right)\)

A= B+C

Ta có: \(B=\dfrac{1}{31}+\dfrac{1}{32}+...+\dfrac{1}{60}>\dfrac{1}{60}+\dfrac{1}{60}+...+\dfrac{1}{60}\)

\(B=\dfrac{1}{31}+\dfrac{1}{32}+...+\dfrac{1}{60}>30.\dfrac{1}{60}=\dfrac{1}{2}\) (1)

Lại có: \(C=\dfrac{1}{61}+\dfrac{1}{62}+...+\dfrac{1}{90}>\dfrac{1}{90}+\dfrac{1}{90}+...+\dfrac{1}{90}\)

\(C=\dfrac{1}{61}+\dfrac{1}{62}+...+\dfrac{1}{90}>30.\dfrac{1}{90}=\dfrac{1}{3}\) (2)

Từ (1) và (2) => \(A>\dfrac{1}{2}+\dfrac{1}{3}=\dfrac{5}{6}\)

Vậy \(A>\dfrac{5}{6}\)