Chứng minh rằng: 1/3 - 2/3^2 +3/3^3 - ... - 2014/3^2014 < 1/5
Cho A=(2014+1).(2014+2).(2014+3)+.....+(2014+2014)A=(2014+1).(2014+2).(2014+3)+.....+(2014+2014)
Chứng minh rằng A chia hết cho 2\(^{2014}\)
Chứng minh rằng: 1/5+2/52+3/53+4/54+...+2014/52014 < 5/8
Chứng minh rằng 1/2^3 + 1/3^3 + 1/4^3 + 1/5^3 + ......... + 1/2014^3 < 1/4
Đặt \(A=\frac{1}{2^3}+\frac{1}{3^3}+\frac{1}{4^3}+...+\frac{1}{2014^3}< B=\frac{1}{1.2.3}+\frac{1}{2.3.4}+\frac{1}{3.4.5}+...+\frac{1}{2013.2014.2015}\)
Mà \(2B=\frac{2}{1.2.3}+\frac{2}{2.3.4}+\frac{2}{3.4.5}+...+\frac{2}{2013.2014.2015}\)
\(=\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+\frac{1}{3.4}-\frac{1}{4.5}+...+\frac{1}{2013.2014}-\frac{1}{2014.2015}\)
\(=\frac{1}{2}-\frac{1}{2014.2015}< \frac{1}{2}\)
\(\Rightarrow B< \frac{1}{4}\)
Vậy \(A< \frac{1}{4}\)
Chứng minh rằng 1+2/2+3/2^2+4/2^3+....+2014/2^2013+2015/2^2014 <4
Cho A=(2014+1)(2014+2)(2014+3)...(2014+2014) . Chứng minh rằng A chia hết cho 22004
Cho A = \(\dfrac{2015}{2014^2+1}+\dfrac{2015}{2014^2+2}+\dfrac{2015}{2014^3+3}+....+\dfrac{2015}{2014^2+2014}\)
Chứng minh rằng A không là số nguyên dương
Các bạn ơi , giúp mình với T T
Chứng minh:
\(\frac{1}{3}-\frac{2}{3^2}+\frac{3}{3^3}-...-\frac{2014}{3^{2014}}< \frac{1}{5}\)
Đặt A = \(\frac{1}{3}-\frac{2}{3^2}+\frac{3}{3^3}-...-\frac{2014}{3^{2014}}\)
3A = \(1-\frac{2}{3}+\frac{3}{3^2}-....-\frac{2014}{3^{2013}}\)
3A + A = \(\left(1-\frac{2}{3}+\frac{3}{3^2}-...-\frac{2014}{3^{2013}}\right)+\left(\frac{1}{3}-\frac{2}{3^2}+\frac{3}{3^3}-....-\frac{2014}{3^{2014}}\right)\)
4A = \(1-\frac{1}{3}+\frac{1}{3^2}-...-\frac{1}{3^{2013}}-\frac{2014}{3^{2014}}\)
=> 4A < \(1-\frac{1}{3}+\frac{1}{3^2}-...-\frac{1}{3^{2013}}\) (1)
Đặt B = \(1-\frac{1}{3}+\frac{1}{3^2}-...-\frac{1}{3^{2013}}\)
3B = \(3-1+\frac{1}{3}-...-\frac{1}{3^{2012}}\)
3B + B = \(\left(3-1+\frac{1}{3}-...-\frac{1}{3^{2012}}\right)+\left(1-\frac{1}{3}+\frac{1}{3^2}-...-\frac{1}{3^{2013}}\right)\)
4B = \(3-\frac{1}{3^{2013}}\)
=> 4B < 3 => B < \(\frac{3}{4}\)(2)
Từ (1)(2) => 4A < B < \(\frac{3}{4}\)=> A < \(\frac{3}{16}\)<\(\frac{1}{5}\)(dpcm)
Cho A= (2014+1) x (2014+2) x (2014+3) x ... x (2014+2014). chứng minh rằng A chia hết cho 2^2014