Question

[ta duc manh]

Cho E=1/3+2/3^2+3/3^3+...+100/3^100

CMR: E<3/4

Answer 1

[Mạch ngọc đức anh]

1/3E=1/3^2+2/3^3+...+100/3^101

E-1/3E=1/3+1/3^2+1/3^3+...+1/3^100-1/3^101

2/3E=1/3+1/3^2+1/3^3+...+1/3^100-1/3^101

Đặt B=1/3+1/3^2+...+1/3^100

      1/3B=1/3^2+1/3^3+...+1/3^101

B-1/3B=1/3-1/3^101

2/3B=1/3-1/3^101

mà 1/3-1/3^101<1/3

=>2/3B<1/3

=>B<1/2

thay B vào E ta có

2/3E=B-1/3^101

Mà B-1/3^101<B

=>2/3E<B

Mà B<1/2

=>2/3E<1/2

=>E<3/4

k cho mk nha