[tex] \frac{1}{1001}>\frac{1}{1002}>...>\frac{1}{2000}[/tex]
=> [tex] \frac{1}{1001}+\frac{1}{1002}+...+\frac{1}{2000}>\frac{1}{2000}+\frac{1}{2000}+...+\frac{1}{2000}[/tex]
( [tex] \frac{1}{2000} [/tex] se repeta de 1000 de ori)
=> [tex] \frac{1}{1001}+\frac{1}{1002}+...+\frac{1}{2000}>\frac{1}{2000}*1000[/tex]
=> [tex] \frac{1}{1001}+\frac{1}{1002}+...+\frac{1}{2000}>\frac{1}{2}[/tex]
=> S> [tex] \frac{1}{2} [/tex] (1)
[tex] \frac{1}{2000}< \frac{1}{1999}<...<\frac{1}{1001}<\frac{1}{1000}[/tex]
=> [tex] \frac{1}{2000}+ \frac{1}{1999}+...+\frac{1}{1001}<\frac{1}{1000}+\frac{1}{1000}+...+\frac{1}{1000}[/tex]
( [tex] \frac{1}{1000}[/tex] se repeta de 1000 de ori)
=> [tex] \frac{1}{2000}+ \frac{1}{1999}+...+\frac{1}{1001}<\frac{1}{1000}*1000[/tex]
=>[tex] \frac{1}{2000}+ \frac{1}{1999}+...+\frac{1}{1001}<1[/tex]
=> S<1 (2)
Din (1) si (2) => [tex] \frac{1}{2} [/tex]
