Các câu hỏi tương tự
Với \(n\inℕ^∗\), cho:
\(A=1+\frac{1}{3}+...+\frac{1}{2n-3}+\frac{1}{2n-1}\)
\(B=\frac{1}{1\left(n-1\right)}+\frac{1}{3\left(2n-3\right)}+...+\frac{1}{\left(2n-3\right)\cdot3}+\frac{1}{\left(2n-1\right)\cdot1}\)
Tính \(\frac{A}{B}\).
Chứng minh: \(\frac{1}{3\left(1+\sqrt{2}\right)}+\frac{1}{5\left(\sqrt{2}+\sqrt{3}\right)}+\frac{1}{7\left(\sqrt{3}+\sqrt{4}\right)}+...+\frac{1}{\left(2n+1\right)\left(\sqrt{n}+\sqrt{n+1}\right)}\)\(< \frac{1}{2}\)
\(\sin^3\frac{x}{3}+3\sin^3\frac{x}{3^2}+...+3^{n-1}\sin^3\frac{x}{3}=\frac{1}{4}\left(3^n\sin^3\frac{x}{3^n}-\sin x\right)\)\(\frac{1}{2}.\frac{3}{4}.\frac{5}{6}...\frac{2n+1}{2n+2}<\frac{1}{\sqrt{3n+4}}\left(n\ge1\right)\)\(\left(n!\right)^2\ge n^2\ge\left(n+1\right)^{n-1}cho\left(n\ge1\right)\)
Chứng minh rằng:
\(\frac{1}{4+1^4}+\frac{3}{4+3^4}+\frac{5}{4+5^4}+....+\frac{2n-1}{4+\left(2n-1\right)^4}=\frac{n^2}{4n^2+1}\)
CMR:
M=\(\frac{1}{3.\left(\sqrt{1}+\sqrt{2}\right)}\)+\(\frac{1}{5.\left(\sqrt{2}+\sqrt{3}\right)}\) +...+\(\frac{1}{\left(2n+1\right).\left(\sqrt{n}+\sqrt{n+1}\right)}< \frac{1}{2}\)
1) \(\frac{1}{2}.\frac{3}{4}.\frac{5}{6}....\frac{\left(2n-1\right)}{2n}\le\frac{1}{\sqrt{3n+1}}\)( n là số nguyên dương)
Chứng minh rằng: \(\frac{1}{2}.\frac{3}{4}.\frac{5}{6}...\frac{\left(2n-1\right)}{2n}\le\frac{1}{\sqrt{3n+1}}\) ( n là số nguyên dương)
Tính tổng:
\(S=\frac{3}{1^2.3}+\frac{5}{\left(1^2+2^2\right).4}+\frac{7}{\left(1^2+2^2+3^2\right).5}+...+\)\(\frac{2n+1}{\left(1^2+2^2+3^2+...+n^2\right).\left(n+2\right)}\)
mấy Box Toán giúp em với
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
\(\frac{1}{3\left(1+\sqrt{2}\right)}+\frac{1}{5\left(\sqrt{2}+\sqrt{3}\right)}+\frac{1}{7\left(\sqrt{3}+\sqrt{4}\right)}+...+\frac{1}{4003\left(\sqrt{2001}+\sqrt{2002}\right)}<\frac{2001}{2003}\)