Question

a)1/2+1/2^2+1/2^3+1/2^4+........+1/2^n <1

b)1/2^2+1/3^2+1/4^2+ ... +1/10^2 <1

Answer 1

a) Gọi A = 1/2+1/2^2+1/2^3+1/2^4+...+1/2^n

Có 2A = 1+ 1/2+ 1/2^2 +1/2^3 + ... + 1/2^n-1

2A-A= 1 - 1/2^n

=> A = 1 - 1/2^n

Có 1 - 1/2^n < 1

=> A<1

Answer 2

Ta có : \(\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{10^2}=\frac{1}{2.2}+\frac{1}{3.3}+...+\frac{1}{10.10}< \frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{9.10}\)

\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{9}-\frac{1}{10}=1-\frac{1}{10}=\frac{9}{10}< 1\left(\text{đpcm}\right)\)

Answer 3

 

1/2^2+1/3^2+...+1/10^2<1/1*2+1/2*3+1/3*4+...+1/9*10

                                         <1/1-1/2+1/2-1/3+1/3-1/4+...+1/9-1/10

                                         <1/1-1/10

                                         <9/10<1

=>1/2^2+1/3^2+...+1/10^2<1