Chứng minh rằng: \(\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\frac{1}{\sqrt{5}+\sqrt{6}}+...+\frac{1}{\sqrt{79}+\sqrt{80}}>4\)
Tính tổng:
S=\(\sqrt{1+\frac{8.1^2-1}{1^2.3^2}}++\sqrt{1+\frac{8.2^2-1}{3^2.5^2}}++\sqrt{1+\frac{8.3^2-1}{5^2.7^2}}+...++\sqrt{1+\frac{8.1009^2-1}{2017^2.2019^2}}\)
Chứng minh rằng
\(5\sqrt{2}< 1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+......+\frac{1}{\sqrt{50}}< 10\sqrt{2}\)
chứng minh rằng: \(P=\frac{1}{2\sqrt{1}}+\frac{1}{3\sqrt{2}}+\frac{1}{4\sqrt{3}}+...+\frac{1}{2008\sqrt{2007}}\)không phải là số nguyên tố
Chứng minh :
\(\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+.....+\frac{1}{\sqrt{n-1}+\sqrt{n}}=\sqrt{n}-1\)
Cho A = \(\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+...+\frac{1}{\sqrt{120}+\sqrt{121}}\)
B = \(1+\frac{1}{\sqrt{2}}+...+\frac{1}{\sqrt{35}}\)
Chưnhs minh rằng: B > A
Chứng minh: \(\frac{1}{2\sqrt{2}+1\sqrt{1}}+\frac{1}{3\sqrt{3}+2\sqrt{2}}+...+\frac{1}{\left(n+1\right)\sqrt{n+1}+n\sqrt{n}}< 1-\frac{1}{\sqrt{n+1}}\)
Tính : \(S=\frac{7}{\sqrt{2.2}+\sqrt{2.3}}+\frac{7}{\sqrt{2.3}+\sqrt{2.4}}+.....+\frac{7}{\sqrt{2.2018}+\sqrt{2.2019}}\)
Chứng minh rằng\(\frac{1}{3+\sqrt{2}}+\frac{1}{3-\sqrt{2}}=\frac{6}{7}\)
Cho A=\(\frac{1}{\sqrt{1}+\sqrt{3}}+\frac{1}{2\left(\sqrt{3}+\sqrt{5}\right)}+\frac{1}{3\left(\sqrt{5}+\sqrt{7}\right)}+...+\frac{1}{40\left(\sqrt{79}+\sqrt{81}\right)}\)
Chứng minh rằng A<\(\frac{8}{9}\)
Giúp mình với, mình đang rối quá