Question

[Lê Ngọc Linh]

CMR :\(\frac{1}{3}+\frac{2}{3^2}+\frac{3}{3^3}+......+\frac{2008}{3^{2008}}<\frac{3}{4}\) Giúp mình với các bạn ơi

Answer 1

[Đinh Thùy Linh]

• Đặt \(S=\frac{1}{3}+\frac{2}{3^2}+\frac{3}{3^3}+...+\frac{2008}{3^{2008}}\)(1)
• Ta có: \(\frac{1}{3}S=\frac{1}{3^2}+\frac{2}{3^3}+\frac{3}{3^4}+...+\frac{2007}{3^{2008}}+\frac{2008}{3^{2009}}\)(2)
• Trừ vế với vế 2 đửng thức (1) và (2) ta có:

\(S-\frac{1}{3}S=\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+\frac{1}{3^4}+...+\frac{1}{3^{2007}}+\frac{1}{3^{2008}}-\frac{2008}{3^{2009}}<\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+\frac{1}{3^4}+...+\frac{1}{3^{2007}}+\frac{1}{3^{2008}}\)(3)

• Đặt \(P=\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+\frac{1}{3^4}+...+\frac{1}{3^{2007}}+\frac{1}{3^{2008}}\)
• \(\left(1-\frac{1}{3}\right)P=\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+\frac{1}{3^4}+...+\frac{1}{3^{2007}}+\frac{1}{3^{2008}}-\left(\frac{1}{3^2}+\frac{1}{3^3}+\frac{1}{3^4}+...+\frac{1}{3^{2008}}+\frac{1}{3^{2009}}\right)=\frac{1}{3}-\frac{1}{3^{2009}}<\frac{1}{3}\)
• \(\frac{2}{3}P<\frac{1}{3}\Rightarrow P<\frac{1}{2}\)(4)
• Từ (3) và (4) 

\(\Rightarrow\frac{2}{3}S<\frac{1}{2}\Rightarrow S<\frac{3}{4}\)(ĐPCM)