Các câu hỏi tương tự
Cmr: \(2\sqrt{n+1}-2\sqrt{n}<\frac{1}{\sqrt{n}}<2\sqrt{n}-2\sqrt{n-1}\)
Từ đó suy ra: \(2004<1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+......+\frac{1}{\sqrt{1006009}}<2005\)
a. Chứng minh: \(\sqrt{n+1}-\sqrt{n}>\frac{1}{2\sqrt{n}+1}\) (với n là số tự nhiên)
b. Chứng minh :\(1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{2005}}< 2.\sqrt{2005}\)
chứng minh rằng \(\frac{1}{\left(n+1\right)\sqrt{n}+n\sqrt{n+1}}=\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\)với \(n\inℕ^∗\)\
Áp dụng tính tổng
\(S=\frac{1}{2\sqrt{1}+1\sqrt{2}}+\frac{1}{3\sqrt{2}+2\sqrt{3}}+...+\frac{1}{400\sqrt{399}+399\sqrt{400}}\)
chứng minh : \(\frac{1}{2\sqrt{2}+1\sqrt{1}}+\frac{1}{3\sqrt{3}+2\sqrt{2}}+...+\frac{1}{\left(n+1\right)\sqrt{n+1}+n\sqrt{n}}< 1-\frac{1}{\sqrt{n+1}}\)
a/Chứng minh rằng \(\frac{2}{\left(2n+1\right)\sqrt{n}+\sqrt{n+1}}=\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\)
b/Áp dụng chứng minh
\(\frac{1}{3\left(1+\sqrt{2}\right)}+\frac{1}{5\left(\sqrt{2}+\sqrt{3}\right)}+\frac{1}{7\left(\sqrt{3}+\sqrt{4}\right)}+...+\frac{1}{4003\left(\sqrt{2001}+\sqrt{2002}\right)}<\frac{2001}{2003}\)
a, Chứng minh
\(\frac{1}{\left(n+1\right).\sqrt{n}+n\sqrt{n+1}}=\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\)
b, Áp dụng
\(S=\frac{1}{2\sqrt{1}+1\sqrt{2}}+\frac{1}{3\sqrt{2}+2\sqrt{3}}+\frac{1}{3\sqrt{4}+4\sqrt{3}}......+\frac{1}{400\sqrt{399}+399\sqrt{400}}\)
cho \(P=\frac{1}{\sqrt{n^2+1}}+\frac{1}{\sqrt{n^2+2}}+\frac{1}{\sqrt{n^2+3}}+...+\frac{1}{\sqrt{n^2+n}}\)
chứng minh rằng P<1
Chứng minh:
\(\frac{1}{2\sqrt{2}+1\sqrt{1}}+\frac{1}{3\sqrt{3}+2\sqrt{2}}+...+\frac{1}{\left(n+1\right)\sqrt{n+1}+n\sqrt{n}}< 1-\frac{1}{\sqrt{n+1}}\left(n\in N\right)\)
Chứng minh rằng
\(\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{4}}+...+\frac{1}{\sqrt{n}}< 2\sqrt{n}-2\)