Question

Chứng minh :

\(\dfrac{1}{2}\) < \(\dfrac{1}{51}+\dfrac{1}{52}+\dfrac{1}{53}+....+\dfrac{1}{100}\) < 1

Answer 1

Ta thấy:

\(\dfrac{1}{51}< \dfrac{1}{50}\)

\(\dfrac{1}{52}< \dfrac{1}{50}\)

...

\(\dfrac{1}{100}< \dfrac{1}{50}\)

\(\Rightarrow\dfrac{1}{51}+\dfrac{1}{52}+...+\dfrac{1}{100}< \dfrac{1}{50}.50=1\)

\(\Rightarrow\dfrac{1}{51}+\dfrac{1}{52}+...+\dfrac{1}{100}< 1\left(1\right)\)

Lại có:

\(\dfrac{1}{51}>\dfrac{1}{100}\)

\(\dfrac{1}{52}>\dfrac{1}{100}\)

...

\(\dfrac{1}{100}=\dfrac{1}{100}\)

\(\Rightarrow\dfrac{1}{51}+\dfrac{1}{52}+...+\dfrac{1}{100}>\dfrac{1}{100}.50=\dfrac{1}{2}\)

\(\Rightarrow\dfrac{1}{51}+\dfrac{1}{52}+...+\dfrac{1}{100}>\dfrac{1}{2}\left(2\right)\)

Từ (1),(2)\(\Rightarrow\)\(\dfrac{1}{2}< \dfrac{1}{51}+\dfrac{1}{52}+...+\dfrac{1}{100}< 1\)