với số nguyên dương lớn hơn 1
a)cmr \(\dfrac{1}{1^2}+\dfrac{1}{2^2}+...+\dfrac{1}{n^2}< 2-\dfrac{1}{n}\)
b)cmr \(\dfrac{1}{1^2}+\dfrac{1}{2^2}+...+\dfrac{1}{n^2}< \dfrac{5}{3}\)
với n số nguyên dương lớn hơn 1
a) cmr \(\dfrac{1}{1^2}+\dfrac{1}{2^2}+...+\dfrac{1}{n^2}< 2-\dfrac{1}{n}\)
b)cmr \(\dfrac{1}{1^2}+\dfrac{1}{2^2}+...+\dfrac{1}{n^2}< \dfrac{5}{3}\)
CMR 1/2+1/3√2+1/4√3+.....+1/(n+1)√... <2
Cmr: S = 1 + \(\sqrt{\frac{2+1}{2}}\)+ \(\sqrt[3]{\frac{3+1}{3}}\)+ ... + \(\sqrt[n]{\frac{n+1}{n}}\)< n + 1 ( Gợi ý: Cmr \(\sqrt[n]{\frac{n+1}{n}}\)< 1+\(\dfrac{1}{k^2}\) )
Cho abc=1
cmr: \(\dfrac{1}{ab+b+1}+\dfrac{1}{bc+c+1}+\dfrac{1}{ca+a+1}=1\)
CMR: \(\frac{\pi}{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+...+\frac{1}{2n-1}-\frac{1}{2n-1}+...\)
Cho \(A_n=\dfrac{1}{\left(2n+1\right)\sqrt{2n-1}},\forall n\in N\text{*}\)
CMR: \(A_1+A_2+...+A_n< 1\)
cho a,b,c là các số thực dương thỏa mãn abc=1 CMR (4a-1)/((2b+1)^2)+(4b-1)/((2c+1)^2)+(4c-1)/((2a+1)^2)>1
a) cho x,y dương. CMR: \(\frac{1}{x}+\frac{1}{y}\ge\frac{4}{x+y}\)
b) cho a+b+c=1 CMR: \(\frac{a}{a+b^2}+\frac{b}{b+c^2}+\frac{c}{c+a^2}\le\frac{1}{4}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\)
CMR với mọi số tự nhiên lớn hơn 2 thì :
\(1+\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{2^n-1}>\dfrac{n}{2}\)