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9 tháng 4 2017
Ta có 1/3+1/4>1/4+1/4=1/2
Suy ra , 1/2+1/3+1/4>1
* 1/5+1/6+1/7+1/8>1/8+1/8+1/8+1/8=4/8=1/2 (1)
*1/9+1/10+1/11+...+1/17>1/17+1/17+1/17+...+1/17(9 p/s1/7)=9/17 >8.5/17=1/2 (2)
Từ (1) và (2) , suy ra : 1/5+1/6+1/7+...+1/17 > 1/2+1/2 = 1
Vậy: 1/2+1/3+1/4+...+1/17 > 2
Mà 2 < 1/2+1/3+1/4+...+1/17 < 1/2+1/3+1/4+...+1/63
Suy ra : 1/2+1/3+1/4+...+1/63 > 2 ( ĐPCM )
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
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
1/2.4+1/4.6+1/6.8+...+1/2012/2014
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