Question

chứng minh luôn luôn tồn tại 1 số tự nhiên n sao cho 1+1/2+1/3+...+1/n>1000

Answer 1

Ta chọn n=2^1999

Ta có:1+1/2+1/3+...+1/n=1+1/2+(1/3+1/2²)+(1/5+1/6+1/7+1/2^3)+(1/9+...+1/2^4)+...+(1/2^199​8+1+...+1/2¹⁹⁹⁹)>1+1/2+1/2².2+1/2³.2²+1/2⁴.2³+...+1/2¹⁹⁹⁹.2¹⁹⁹⁸=1+1/2.1999=1000,5>1000(đpcm)

Answer 2

Số đó là 2^1999

Answer 3

Ta chọn n=2^1999 Ta có:1+1/2+1/3+...+1/n=1+1/2+(1/3+1/22)+(1/5+1/6+1/7+1/2^3)+(1/9+...+1/2^4)+...+(1/2199​8+1+...+1/21999)>1+1/2+1/22.2+1/23.22+1/24.23+...+1/21999.21998=1+1/2.1999=1000,5>1000 sr a>1000