Question

Chứng tỏ rằng:

\(1\)+\(\frac{1}{2}\)+\(\frac{1}{3}\)+...+\(\frac{1}{62}\)+\(\frac{1}{63}\)+\(\frac{1}{64}\)\(>\)\(4\)

Answer 1

1 + 1/2 + 1/3 + ... + 1/62 + 1/63 + 1/64

= 1 + 1/2 + (1/3 + 1/4) + (1/5 + 1/6 + 1/7 + 1/8) + (1/9 + 1/10 + ... + 1/16) + (1/17 + 1/18 + ... + 1/32) + (1/33 + 1/34 + ... + 1/64) 

> 1 + 1/2 + 1/4 × 2 + 1/8 × 4 + 1/16 × 8 + 1/32 × 16 + 1/64 × 32

> 1 + 1/2 + 1/2 + 1/2 + 1/2 + 1/2 + 1/2

> 1 + 1/2 × 6

> 1 + 3

> 4

Answer 2

1 + 1/2 + 1/3 + ... + 1/62 + 1/63 + 1/64

= 1 + 1/2 + (1/3 + 1/4) + (1/5 + 1/6 + 1/7 + 1/8) + (1/9 + 1/10 + ... + 1/16) + (1/17 + 1/18 + ... + 1/32) + (1/33 + 1/34 + ... + 1/64) 

> 1 + 1/2 + 1/4 × 2 + 1/8 × 4 + 1/16 × 8 + 1/32 × 16 + 1/64 × 32

> 1 + 1/2 + 1/2 + 1/2 + 1/2 + 1/2 + 1/2

> 1 + 1/2 × 6

> 1 + 3

> 4

Answer 3

1/2.4+1/4.6+1/6.8+...+1/2012/2014

Answer 4

Ta có: \(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{62}+\frac{1}{63}+\frac{1}{64}\)

\(=1+\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\right)+\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\right)+\left(\frac{1}{9}+\frac{1}{10}+...+\frac{1}{16}\right)+...+\)

\(\left(\frac{1}{17}+\frac{1}{18}+...+\frac{1}{32}\right)+\left(\frac{1}{33}+\frac{1}{34}+...+\frac{1}{64}\right)\)

Ta có: \(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}>\frac{1}{2}+\frac{1}{4}+\frac{1}{4}=1\)

\(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}>\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}=\frac{1}{8}.4=\frac{1}{2}\)

\(\frac{1}{9}+\frac{1}{10}+...+\frac{1}{16}>\frac{1}{16}+\frac{1}{16}+...+\frac{1}{16}=\frac{1}{16}.8=\frac{1}{2}\)

\(\frac{1}{33}+\frac{1}{34}+...+\frac{1}{64}>\frac{1}{64}+\frac{1}{64}+...+\frac{1}{64}=\frac{1}{64}.32=\frac{1}{2}\)

=> Biểu thức > 4