Question

[bui thi hue]

chứng tỏ rằng: 1/501 + 1/502 + 1/503 + ... + 1/1000 < 1

Answer 1

[bảo nam trần]

Ta có: \(\dfrac{1}{501}< \dfrac{1}{500}\)

\(\dfrac{1}{502}< \dfrac{1}{500}\)

\(\dfrac{1}{503}< \dfrac{1}{500}\)

..................

\(\dfrac{1}{1000}< \dfrac{1}{500}\)

\(\Rightarrow\dfrac{1}{501}+\dfrac{1}{502}+\dfrac{1}{503}+...+\dfrac{1}{1000}< \dfrac{1}{500}+\dfrac{1}{500}+\dfrac{1}{500}+...+\dfrac{1}{500}\)

\(\Rightarrow\dfrac{1}{501}+\dfrac{1}{502}+\dfrac{1}{503}+...+\dfrac{1}{1000}< \dfrac{500}{500}=1\)

Vậy \(\dfrac{1}{501}+\dfrac{1}{502}+\dfrac{1}{503}+...+\dfrac{1}{1000}< 1\)

Answer 2

[Minh Anh Đặng Thị]

Đặt A = \(\dfrac{1}{501}+\dfrac{1}{502}+\dfrac{1}{503}+...+\dfrac{1}{1000}\)

Ta thấy A có 500 phân số.

Ta có: \(\dfrac{1}{501}< \dfrac{1}{500}\\ \dfrac{1}{502}< \dfrac{1}{500}\)

....................

\(\dfrac{1}{1000}< \dfrac{1}{500}\)

\(\Rightarrow\) A< \(\dfrac{1}{500}+\dfrac{1}{500}+...+\dfrac{1}{500}\)( có 500 phân số \(\dfrac{1}{500}\))

\(\Rightarrow A< 500.\dfrac{1}{500}\\ \Rightarrow A< \dfrac{500}{500}\\ \Rightarrow A< 1\)

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