\(M=\dfrac{n^3+2n^2-1}{n^3+2n^2+2n+1}\)
\(=\dfrac{n^3+n^2+n^2+n-n-1}{\left(n+1\right).\left(n^2-n+1\right)+2n.\left(n+1\right)}\)
\(=\dfrac{n^2\left(n+1\right)+n\left(n-1\right)-\left(n+1\right)}{\left(n+1\right).\left(n^2-n+1+2n\right)}\)
\(=\dfrac{\left(n+1\right).\left(n^2+n-1\right)}{\left(n+1\right).\left(n^2+n+1\right)}\)
\(=\dfrac{n^2+n-1}{n^2+n+1}\)
C/m với mọi n nguyên dương thì
\(\dfrac{1}{2\sqrt{2}}+\dfrac{1}{4\sqrt{3}}+.....+\dfrac{1}{2n\sqrt{n+1}}+\dfrac{1}{\sqrt{n+1}}>1\)
cmr:
\(\dfrac{1}{2}.\dfrac{3}{4}.\dfrac{5}{6}....\dfrac{2n-1}{2n}\le\dfrac{1}{\sqrt{3n+1}}\left(\forall n\ge1\right)\)
Cho các số thực a,b,x,y thõa mãn \(x^2+y^2=1,\dfrac{x^4}{a}+\dfrac{y^4}{b}=\dfrac{1}{a+b}\).
C/m : \(\dfrac{x^{2n}}{a^n}+\dfrac{y^{2n}}{b^n}=\dfrac{2}{\left(a+b\right)^n},\forall n\in N\)
Chứng minh \(\dfrac{1}{2\sqrt{2}}+\dfrac{1}{4\sqrt{3}}+\dfrac{1}{6\sqrt{4}}+....+\dfrac{1}{2n\sqrt{n+1}}+\dfrac{1}{\sqrt{n+1}}>1\)
Em không rõ là > hay < 1 ấy ạ. Anh chị nào giúp em với
cho \(a_{1}=\dfrac{1}{2}, a_{n+1} = (\dfrac{2n-1}{2n+2}). a_{n}\) với mọi số nguyên a không vượt quá 2005. CMR \(a_{1} + a_{2}+......+a_{2006}<1 \)
cho dãy số:
\(a_1=1,a_2=1+\dfrac{1}{3},...,a_n=1+\dfrac{1}{3}+\dfrac{1}{5}+...+\dfrac{1}{2n-1}\)
cmr:\(\dfrac{1}{a_1^2}+\dfrac{1}{3a_2^2}+...+\dfrac{1}{\left(2n-1\right)a_n^2}< 2\)
cho dãy số :\(a_1=1,a_2=1+\dfrac{1}{3},.....,a_n=1+\dfrac{1}{3}+\dfrac{1}{5}+...+\dfrac{1}{2n-1}\)
cmr:
\(\dfrac{1}{a_1^2}+\dfrac{1}{3a_2^2}+...+\dfrac{1}{\left(2n-1\right)a_n^2}< 2\)
Tính các tổng sau:
A=\(\dfrac{1}{2.5}+\dfrac{1}{5.8}+\dfrac{1}{8.11}+.....+\dfrac{1}{\left(3n-1\right)\left(3n+2\right)}\)
B=\(\dfrac{1}{1.3.5}+\dfrac{1}{3.5.7}+\dfrac{1}{5.7.9}+....+\dfrac{1}{\left(2n-1\right)\left(2n+1\right)\left(2n+3\right)}\)
C=\(\sqrt{1+\dfrac{1}{1^2}+\dfrac{1}{2^2}}+\sqrt{1+\dfrac{1}{2^2}+\dfrac{1}{3^2}}+\sqrt{1+\dfrac{1}{3^2}+\dfrac{1}{4^2}}+....+\sqrt{1+\dfrac{1}{2018^2}+\dfrac{1}{2019^2}}\)
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: \(S_n< \dfrac{1}{2}\)
