Hãy nhập câu hỏi của bạn vào đây, nếu là tài khoản VIP, bạn sẽ được ưu tiên trả lời.
\(E=\dfrac{1}{3}-\dfrac{2}{3^2}+\dfrac{3}{3^3}-\dfrac{4}{3^4}+...+\dfrac{2015}{3^{2015}}-\dfrac{2016}{3^{2016}}\\ 3E=1-\dfrac{2}{3}+\dfrac{3}{3^2}-\dfrac{4}{3^3}+...+\dfrac{2015}{3^{2014}}-\dfrac{2016}{3^{2015}}\\ 3E+E=\left(1-\dfrac{2}{3}+\dfrac{3}{3^2}-\dfrac{4}{3^3}+...+\dfrac{2015}{3^{2014}}-\dfrac{2016}{3^{2015}}\right)+\left(\dfrac{1}{3}-\dfrac{2}{3^2}+\dfrac{3}{3^3}-\dfrac{4}{3^4}+...+\dfrac{2015}{3^{2015}}-\dfrac{2016}{3^{2016}}\right)\\ 4E=1-\dfrac{1}{3}+\dfrac{1}{3^2}-\dfrac{1}{3^3}+...+\dfrac{1}{3^{2014}}-\dfrac{1}{3^{2015}}-\dfrac{2016}{3^{2016}}\\ 12E=3-1+\dfrac{1}{3}-\dfrac{1}{3^2}+...+\dfrac{1}{3^{2013}}-\dfrac{1}{3^{2014}}-\dfrac{6048}{3^{2016}}\\ 4E+12E=\left(3-1+\dfrac{1}{3}-\dfrac{1}{3^2}+...+\dfrac{1}{3^{2013}}-\dfrac{1}{3^{2014}}-\dfrac{2016}{3^{2015}}\right)+\left(1-\dfrac{1}{3}+\dfrac{1}{3^2}-\dfrac{1}{3^3}+...+\dfrac{1}{3^{2014}}-\dfrac{1}{3^{2015}}-\dfrac{2016}{3^{2016}}\right)\\ 16E=3-\dfrac{2017}{3^{2015}}-\dfrac{2016}{3^{2016}}\\ 16E=3-\left(\dfrac{2017}{3^{2015}}+\dfrac{672}{3^{2015}}\right)\\ 16E=3-\dfrac{2689}{3^{2015}}< 3\\ \Rightarrow E< \dfrac{3}{16}\)
\(E=\dfrac{1}{3}-\dfrac{2}{3^2}+\dfrac{3}{3^3}-\dfrac{4}{3^4}+...+\dfrac{2015}{3^{2015}}-\dfrac{2016}{3^{2016}}\\ 3E=1-\dfrac{2}{3}+\dfrac{3}{3^2}-\dfrac{4}{3^3}+...+\dfrac{2015}{3^{2014}}-\dfrac{2016}{3^{2015}}\\ 3E+E=\left(1-\dfrac{2}{3}+\dfrac{3}{3^2}-\dfrac{4}{3^3}+...+\dfrac{2015}{3^{2014}}-\dfrac{2016}{3^{2015}}\right)+\left(\dfrac{1}{3}-\dfrac{2}{3^2}+\dfrac{3}{3^3}-\dfrac{4}{3^4}+...+\dfrac{2015}{3^{2015}}-\dfrac{2016}{3^{2016}}\right)\\ 4E=1-\dfrac{1}{3}+\dfrac{1}{3^2}-\dfrac{1}{3^3}+...+\dfrac{1}{3^{2014}}-\dfrac{1}{3^{2015}}-\dfrac{2016}{3^{2016}}\\ 4E< 1-\dfrac{1}{3}+\dfrac{1}{3^2}-\dfrac{1}{3^3}+...+\dfrac{1}{3^{2014}}-\dfrac{1}{3^{2015}}\left(1\right)\)
Gọi \(D=1-\dfrac{1}{3}+\dfrac{1}{3^2}-\dfrac{1}{3^3}+...-\dfrac{1}{3^{2015}}\)
\(3D=3-1+\dfrac{1}{3}-\dfrac{1}{3^2}+...+\dfrac{1}{3^{2013}}-\dfrac{1}{3^{2014}}\\ 3D+D=\left(3-1+\dfrac{1}{3}-\dfrac{1}{3^2}+...+\dfrac{1}{3^{2013}}-\dfrac{1}{3^{2014}}\right)+\left(1-\dfrac{1}{3}+\dfrac{1}{3^2}-\dfrac{1}{3^3}+...+\dfrac{1}{3^{2014}}-\dfrac{1}{3^{2015}}\right)\\ 4D=3-\dfrac{1}{3^{2015}}< 3\\ \Rightarrow D< \dfrac{3}{4}\left(2\right)\)
Từ (1) và (2) ta có:
\(4E< \dfrac{3}{4}\\ \Rightarrow E< \dfrac{3}{16}\)
\(A=1+3+3^2+3^3+3^4+...+3^{2015}\)
\(=\left(1+3+3^2+3^3\right)+\left(3^4+3^5+3^6+3^7\right)+...+\left(3^{2012}+3^{2013}+3^{2014}+3^{2015}\right)\)
\(=\left(1+3+3^2+3^3\right)+3^4\left(1+3+3^2+3^3\right)+...+3^{2012}\left(1+3+3^2+3^3\right)\)
\(=\left(1+3+3^2+3^3\right)\left(1+3^4+...+3^{2012}\right)\)
\(=40\left(1+3^4+...+3^{2012}\right)\)\(⋮\)\(5\)
\(B=2+2^2+2^3+...+2^{2016}\)
\(=\left(2+2^2+2^3+2^4\right)+\left(2^5+2^6+2^7+2^8\right)+...+\left(2^{2013}+2^{2014}+2^{2015}+2^{2016}\right)\)
\(=2\left(1+2+2^2+2^3\right)+2^5\left(1+2+2^2+2^3\right)+..+2^{2013}\left(1+2+2^2+2^3\right)\)
\(=\left(1+2+2^2+2^3\right)\left(2+2^5+...+2^{2013}\right)\)
\(=15\left(2+2^5+...+2^{2013}\right)\)\(⋮\)\(15\)