Question

So sánh tổng S= \(\dfrac{1}{2}\)+\(\dfrac{2}{2^2}\)+\(\dfrac{3}{2^3}\)+...+\(\dfrac{n}{2^n}\)+...+\(\dfrac{2017}{2^{2017}}\)với 2 (\(n\in N\)*)

Answer 1

Giải:

\(S=\dfrac{1}{2}+\dfrac{2}{2^2}+...+\dfrac{n}{2^n}+...+\dfrac{2017}{2^{2017}}\) 

Với \(n>2\) thì \(\dfrac{n}{2^n}=\dfrac{n+1}{2^{n-1}}-\dfrac{n+2}{2^n}\) 

Ta có:

\(\dfrac{n+1}{2^{n-1}}=\dfrac{n+1}{2^n:2}=\dfrac{2.\left(n+1\right)}{2^n}\) 

\(\Rightarrow\dfrac{n+1}{2^{n-1}}-\dfrac{n+2}{2^n}\) 

\(=\dfrac{2.\left(n+1\right)}{2^n}-\dfrac{n+2}{2^n}\) 

\(=\dfrac{2.\left(n+1\right)-n-2}{2^n}\) 

\(=\dfrac{n}{2^n}\) 

  \(\Leftrightarrow S=\dfrac{1}{2}+\left(\dfrac{2+1}{2^{2-1}}-\dfrac{2+2}{2^2}\right)+...+\left(\dfrac{2016+1}{2^{2015}}-\dfrac{2018}{2^{2016}}\right)+\left(\dfrac{2017+1}{2^{2016}}-\dfrac{2019}{2^{2017}}\right)\)

\(S=\dfrac{1}{2}+\dfrac{3}{2}+\dfrac{2019}{2017}\) 

\(S=2-\dfrac{2019}{2017}\)  

\(\Leftrightarrow S=2-\dfrac{2019}{2017}< 2\) 

Hay \(S< 2\)