Question

Докажите что 1/2<1/1001+1/1002+...+1/2000<1

Answer 1

1-(1/2)+(1/3)-(1/4)+...+(1/1999)-(1/2000) =
= (1/1001)+(1/1002)+...+(1/2000).

Answer 2

$frac{1}{1001}+frac{1}{1002}+...+frac{1}{2000}>frac{1}{2000}+frac{1}{2000}+...+frac{1}{2000}=1000*frac{1}{2000}=frac{1}{2};$

 

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