Chứng minh đẳng thức sau \(\frac{1}{1.2.3}+\frac{1}{2.3.4}+\frac{1}{3.4.5}+...+\frac{1}{\left(n-1\right)n\left(n+1\right)}=\frac{\left(n-1\right)\left(n+2\right)}{4n\left(n+1\right)}\) với \(n\ge2\)
Chứng minh các đẳng thức thức sau với số tự nhiên n>= 1, tùy ý
a)1+2+3+...+n=\(\frac{n\left(n+1\right)}{2}\)
b)\(1^2+2^2+3^2+...+n^2=\frac{n\left(n+1\right)\left(2n+1\right)}{6}\)
c)\(1^3+2^3+3^3+...+n^3=\frac{n^2\left(n+1\right)^2}{4}\)
d)1.2.3+...+n(n+1)(n+2)=\(\frac{n\left(n+1\right)\left(n+2\right)\left(n+3\right)}{4}\)
Chứng minh: \(\frac{3}{\left(1x2\right)}+\frac{5}{\left(2x3\right)}+...+\frac{2n+1}{\left(n\left(n+1\right)\right)^2}=\frac{n\left(n+2\right)}{\left(n+1\right)^2}\)
a,Chứng minh với mọi n nguyên dương ta có
\(\frac{n+1}{n^2+1}\)>\(\frac{n+2}{\left(n+1\right)^2+1}\)
b,Chứng minh
0,33<\(\frac{99}{100^2+1}\)+\(\frac{100}{101^2+1}\)+...+\(\frac{148}{149^2+1}\)<0,5
Chứng minh :
a) \(\frac{1}{\left(n-1\right)n\left(n+1\right)}=\frac{1}{2}\left(\frac{1}{\left(n-1\right)n}-\frac{1}{n\left(n+1\right)}\right)\)
Chứng minh rằng :
A= \(\left(1-\frac{3}{2.4}\right).\left(1-\frac{3}{3.5}\right)...\left(1-\frac{3}{n\left(n+2\right)}\right)>\frac{1}{4}\)
\(n\in N;n\ge2\)
help me! (ngu toàn tập)
a)\(\frac{3}{\left(1.2\right)^2}+\frac{5}{\left(2.3\right)^2}+...+\frac{2n+1}{\left[n\left(n+1\right)\right]^2}\)
b)\(\left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)\left(1-\frac{1}{4^2}\right)....\left(1-\frac{1}{n^2}\right)\)
c)\(\frac{150}{5.8}+\frac{150}{8.11}+\frac{150}{11.14}+...+\frac{150}{47.50}\)
d)\(\frac{1}{1.2.3}+\frac{1}{2.3.4}+\frac{1}{3.4.5}+...+\frac{1}{\left(n-1\right)n\left(n+1\right)}\)
\(\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{n\left(n+1\right)\left(n+2\right)}\)
Tìm tích
\(A=\left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)...\left(1-\frac{1}{n^2}\right)\) với n thuộc N và n>2