Violympic toán 9
Các câu hỏi tương tự
1) Chứng minh rằng: 1+dfrac{1}{2sqrt{2}}+dfrac{1}{3sqrt{3}}+...+dfrac{1}{nsqrt{n}} 2sqrt{2}left(nin Nright)
2) Chứng minh rằng: dfrac{2}{3}+sqrt{n+1} 1+sqrt{2}+sqrt{3}+...+sqrt{n} dfrac{2}{3}left(n+1right)sqrt{n}
3) 2sqrt{n}-3 dfrac{1}{sqrt{2}}+dfrac{1}{sqrt{3}}+...+dfrac{1}{sqrt{n}} 2sqrt{n}-2
4) dfrac{sqrt{2}-sqrt{1}}{2+1}+dfrac{sqrt{3}-sqrt{2}}{3+2}+...+dfrac{sqrt{n+1}-sqrt{n}}{n+1+n} dfrac{1}{2}left(1-dfrac{1}{sqrt{n+1}}right)
Đọc tiếp
1) Chứng minh rằng: \(1+\dfrac{1}{2\sqrt{2}}+\dfrac{1}{3\sqrt{3}}+...+\dfrac{1}{n\sqrt{n}}< 2\sqrt{2}\left(n\in N\right)\)
2) Chứng minh rằng: \(\dfrac{2}{3}+\sqrt{n+1}< 1+\sqrt{2}+\sqrt{3}+...+\sqrt{n}< \dfrac{2}{3}\left(n+1\right)\sqrt{n}\)
3) \(2\sqrt{n}-3< \dfrac{1}{\sqrt{2}}+\dfrac{1}{\sqrt{3}}+...+\dfrac{1}{\sqrt{n}}< 2\sqrt{n}-2\)
4) \(\dfrac{\sqrt{2}-\sqrt{1}}{2+1}+\dfrac{\sqrt{3}-\sqrt{2}}{3+2}+...+\dfrac{\sqrt{n+1}-\sqrt{n}}{n+1+n}< \dfrac{1}{2}\left(1-\dfrac{1}{\sqrt{n+1}}\right)\)
Với số tự nhiên n, \(n\ge3\). Đặt \(S_n=\dfrac{1}{3\left(1+\sqrt{2}\right)}+\dfrac{1}{5\left(\sqrt{2}+\sqrt{3}\right)}+...+\dfrac{1}{\left(2n+1\right)\left(\sqrt{n}+\sqrt{n+1}\right)}\). Chứng minh: \(S_n< \dfrac{1}{2}\)
Chứng minh rằng với mọi số nguyên dương n ta đều có \(\dfrac{1}{2\sqrt{1}}+\dfrac{1}{3\sqrt{2}}+\dfrac{1}{4\sqrt{3}}+\dfrac{1}{5\sqrt{4}}+...+\dfrac{1}{\left(n+1\right)\sqrt{n}}< 2\)
Chứng minh rằng:
\(\dfrac{1}{2\sqrt{1}}+\dfrac{1}{3\sqrt{2}}+....+\dfrac{1}{\left(n+1\right)\sqrt{n}}< 2\)
CMR:
\(\dfrac{1}{2\sqrt{1}+1\sqrt{2}}+\dfrac{1}{3\sqrt{2}+2\sqrt{3}}+\dfrac{1}{4\sqrt{3}+3\sqrt{3}}+....+\dfrac{1}{\left(n+1\right)\left(\sqrt{n}+n\sqrt{n+1}\right)}< 1\)
Chứng minh rằng: \(\dfrac{1}{2\sqrt{1}}+\dfrac{1}{3\sqrt{1}}+...+\dfrac{1}{\left(n+1\right)\sqrt{n}}< 2\) với mọi n là số tự nhiên khác 0
10 tháng 10 2021 lúc 10:55
Rút gọn:
\(\sqrt{1+\dfrac{1}{11^2}+\dfrac{1}{12^2}}+\sqrt{1+\dfrac{1}{12^2}+\dfrac{1}{13^2}}+...+1\sqrt{1+\dfrac{1}{\left(n+1\right)^2}+\dfrac{1}{\left(n+2\right)^2}}\)
Chứng minh rằng: \(\left|\dfrac{m}{n}-\sqrt{2}\right|\ge\dfrac{1}{n^2\left(\sqrt{3}+\sqrt{2}\right)}\)
Cho a, b, c > 0 thoả mãn: \(a+b+c=\sqrt{a}+\sqrt{b}+\sqrt{c}=2\). Chứng minh rằng: \(\dfrac{\sqrt{a}}{a+1}+\dfrac{\sqrt{b}}{b+1}+\dfrac{\sqrt{c}}{c+1}=\dfrac{2}{\sqrt{\left(a+1\right)\left(b+1\right)\left(c+1\right)}}\)
CMR:\(1+\dfrac{1}{2\sqrt{1}}+\dfrac{1}{3\sqrt{2}}+...+\dfrac{1}{\left(n+1\right)\sqrt{n}}< 2\)