Question

so sánh ;

A=\(\frac{1}{5^1}+\frac{1}{5^2}+....+\frac{1}{5^{2014}}+\frac{1}{5^{2015}}\)với \(\frac{1}{4}\)

Answer 1

\(A=\frac{1}{5}+\frac{1}{5^2}+...+\frac{1}{5^{2015}}\)

=>\(5A=1+\frac{1}{5}+...+\frac{1}{5^{2014}}\)

=>\(5A-A=\left(1+\frac{1}{5}+...+\frac{1}{5^{2014}}\right)-\left(\frac{1}{5}+\frac{1}{5^2}+...+\frac{1}{5^{2015}}\right)\)

=>\(4A=1-\frac{1}{5^{2015}}\)

=>\(A=\frac{1-\frac{1}{5^{2015}}}{4}\)

Dễ thấy \(1-\frac{1}{5^{2015}}< 1\Rightarrow\frac{1-\frac{1}{5^{2015}}}{4}< \frac{1}{4}\Rightarrow A< \frac{1}{4}\)

Answer 2

A<1/4