Question

Cho tổng gồm 2014 số hạng:      S= 1/4 + 2/4² + 3/4³ + 4/4⁴ + ... + 2014/4²⁰¹⁴

Chứng mih rằng: S < 1

Answer 1

=>  \(4.S=1+\frac{2}{4}+\frac{3}{4^2}+\frac{4}{4^3}+...+\frac{2014}{4^{2013}}\)

=> 4.S - S = \(\left(1+\frac{2}{4}+\frac{3}{4^2}+\frac{4}{4^3}+...+\frac{2014}{4^{2013}}\right)-\left(\frac{1}{4}+\frac{2}{4^2}+\frac{3}{4^3}+...+\frac{2014}{4^{2014}}\right)\)

=> 3.S = \(=1+\left(\frac{2}{4}-\frac{1}{4}\right)+\left(\frac{3}{4^2}-\frac{2}{4^2}\right)+\left(\frac{4}{4^3}-\frac{3}{4^3}\right)+...+\left(\frac{2014}{4^{2013}}-\frac{2013}{4^{2013}}\right)-\frac{2014}{4^{2014}}\)

=> 3.S =  \(1+\frac{1}{4}+\frac{1}{4^2}+...+\frac{1}{4^{2013}}-\frac{2014}{4^{2014}}\)

Tính A= \(1+\frac{1}{4}+\frac{1}{4^2}+...+\frac{1}{4^{2013}}\)

=> \(4.A=4+1+\frac{1}{4}+\frac{1}{4^2}+...+\frac{1}{4^{2012}}\)

=> 4.A - A = \(4-\frac{1}{4^{2013}}\)=> A= \(\frac{4}{3}-\frac{1}{3.4^{2013}}\)

=> 3.S = \(\frac{4}{3}-\frac{1}{3.4^{2013}}-\frac{2014}{4^{2014}}\) => S = \(\frac{4}{9}-\frac{1}{9.4^{2013}}-\frac{2014}{4^{2014}}

Answer 2

Nếu là 1/2 thì ta so sánh 4/9 < 4/8 = 1/2 => S < 1/2

Answer 3

Sao an loan luc lam S luc lam A vay? Do hoi chang?

Answer 4

Luong Huyen Trang ơi, cái đó là đặt đấy