\(=\frac{1}{2}-\frac{1}{2000}=\frac{999}{2000}\)
Dạng tổng quát :
\(\frac{1}{2.3}+\frac{1}{3.4}+........+\frac{1}{n\left(n+1\right)}=\frac{1}{2}-\frac{1}{n+1}=\frac{n+1-2}{2\left(n+1\right)}=\frac{n-1}{2\left(n+1\right)}\)
\(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{1999.2000}\)
\(=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{1999}-\frac{1}{2000}\)
\(=\frac{1}{2}-\frac{1}{2000}\)
\(=\frac{499}{2000}\)
\(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{1999.2000}=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{1999}-\frac{1}{2000}\)
\(=\frac{1}{2}-\frac{1}{2000}\)
\(=\frac{999}{2000}\)
Study well ! >_<
\(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{1999.2000}\)
\(=\frac{3-2}{2.3}+\frac{4-3}{3.4}+\frac{5-4}{4.5}+...+\frac{2000-1999}{1999.2000}\)
\(=\frac{2}{2.3}-\frac{3}{2.3}+\frac{3}{3.4}-\frac{4}{3.4}+\frac{4}{4.5}-\frac{5}{5.4}+...+\frac{1999}{1999.2000}-\frac{2000}{1999.2000}\)
\(=\frac{1}{3}-\frac{1}{1999}\)
\(=\frac{1999}{5997}+\frac{3}{5997}\)
\(=\frac{2002}{5997}\)
#Thiên_Hy
