Question

Cho n là số nguyên dương. C/m:

\(\dfrac{1}{3}+\dfrac{2}{3^2}+\dfrac{3}{3^3}+...+\dfrac{n}{3^n}< \dfrac{3}{4}\)

Answer 1

Đặt \(A=\dfrac{1}{3}+\dfrac{2}{3^2}+\dfrac{3}{3^3}+...+\dfrac{n}{3^n}\)

\(3A=1+\dfrac{2}{3}+\dfrac{3}{3^2}+...+\dfrac{n}{3^{n-1}}\)

\(\Rightarrow2A=1+\dfrac{1}{3}+\dfrac{1}{3^2}+...+\dfrac{1}{3^{n-1}}-\dfrac{n}{3^n}< 1+\dfrac{1}{3}+\dfrac{1}{3^2}+...+\dfrac{1}{3^{n-1}}\)

Đặt \(B=1+\dfrac{1}{3}+\dfrac{1}{3^2}+...+\dfrac{1}{3^{n-1}}\)

Tương tự ta được \(2B=3-\dfrac{1}{3^{n-1}}< 3\)

\(\Rightarrow B< \dfrac{3}{2}\Rightarrow2A< \dfrac{3}{2}\Rightarrow A< \dfrac{3}{4}\)(đpcm)

Answer 2

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