Chứng minh
1/ 1^2 + 1/2^2 + ... + 1/ 2014 ^2 < 1 3/ 4
Cho C = 1*2*3*.....*2014*(1+1/2+1/3+1/4+.....1/2014) chứng minh C là 1 số tự nhiên chia hết cho 2^2014
Chứng minh: (1/4+2/4^2+3/4^34/4^4+........+2014/4^2014)<1/2
cho P=1+1/2+1/3+1/4+...+1/2^2014 - 1
chứng minh rằng P<2014
Chứng minh:\(S=\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+........+\frac{1}{2014^2}< \frac{2013}{2014}\)
Ta có: \(\frac{1}{2^2}=\frac{1}{2.2}< \frac{1}{1.2}\)
Tương tự : \(\frac{1}{3^2}< \frac{1}{2.3}\); \(\frac{1}{4^2}< \frac{1}{3.4}\); ......... ; \(\frac{1}{2014^2}< \frac{1}{2013.2014}\)
\(\Rightarrow S< \frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+........+\frac{1}{2013.2014}\)
\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+.........+\frac{1}{2013}-\frac{1}{2014}\)
\(=1-\frac{1}{2014}=\frac{2013}{2014}\)
\(\Rightarrow S< \frac{2013}{2014}\left(đpcm\right)\)
cho P=1+1/2+1/3+1/4+...+1/2^2014 - 1
chứng minh rằng P<2014
Chứng minh
1/1!+1/2!+1/3!+1/4!+...+1/2014!<2
Cho S = 1/4+2/42+3/43+...+2014/42014
Chứng minh S < 1/2
Chứng minh rằng:
\(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{2014^2}< \frac{2013}{2014}\)
gọi dãy số trên là A
ta có A<\(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{2013.2014}\)
A<1-\(\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2013}-\frac{1}{2014}\)
A<1-\(\frac{1}{2014}\)=\(\frac{2013}{2014}\)
Vậy A < \(\frac{2013}{2014}\)
Chứng minh : 1/2-1/3+1/4-1/5+...+1/2013-1/2014 < 2/5