Question

[Lê duy anh]

Chứng minh 1/4<1/4+1/4²+1/4³+....+1/4²⁰¹⁵<1/3

Answer 1

[Ác Mộng]

Ta có:A=\(\frac{1}{4}+\frac{1}{4^2}+...+\frac{1}{4^{2015}}\)

=>4A=\(1+\frac{1}{4}+...+\frac{1}{4^{2014}}\)

=>4A-A=(\(1+\frac{1}{4}+...+\frac{1}{4^{2014}}\))-(\(\frac{1}{4}+\frac{1}{4^2}+...+\frac{1}{4^{2015}}\))

=>3A=1-\(\frac{1}{4^{2015}}\)

=>A=\(\frac{1}{3}-\frac{1}{3.4^{2015}}<\frac{1}{3}\)

Do A=\(\frac{1}{4}+\frac{1}{4^2}+...+\frac{1}{4^{2015}}>\frac{1}{4}\)

Vậy \(\frac{1}{4}<\)\(\frac{1}{4}+\frac{1}{4^2}+...+\frac{1}{4^{2015}}\)\(<\frac{1}{3}\)