Question

Va rog, puţin ajutor! :(
S = 1/1001 + 1/1002 + ... + 1/2000. Aratati ca 1/2 < S < 1.
Multumesc! :)

Answer 1

$\frac{1}{1001}>\frac{1}{1002}>...>\frac{1}{2000}$

=> $\frac{1}{1001}+\frac{1}{1002}+...+\frac{1}{2000}>\frac{1}{2000}+\frac{1}{2000}+...+\frac{1}{2000}$

( $\frac{1}{2000}$ se repeta de 1000 de ori)

=> $\frac{1}{1001}+\frac{1}{1002}+...+\frac{1}{2000}>\frac{1}{2000}*1000$

=> $\frac{1}{1001}+\frac{1}{1002}+...+\frac{1}{2000}>\frac{1}{2}$

=> S> $\frac{1}{2}$ (1)

$\frac{1}{2000}< \frac{1}{1999}<...<\frac{1}{1001}<\frac{1}{1000}$


=> $\frac{1}{2000}+ \frac{1}{1999}+...+\frac{1}{1001}<\frac{1}{1000}+\frac{1}{1000}+...+\frac{1}{1000}$


( $\frac{1}{1000}$ se repeta de 1000 de ori)

=> $\frac{1}{2000}+ \frac{1}{1999}+...+\frac{1}{1001}<\frac{1}{1000}*1000$

=>$\frac{1}{2000}+ \frac{1}{1999}+...+\frac{1}{1001}<1$

=> S<1 (2)

Din (1) si (2) => $\frac{1}{2}$