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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
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 )
1/2+1/3+1/4+...+1/63>1/31+ 1/31+...+1/31(62 số hang 1/31)
hay 1/2+1/3+1/4+...+1/63 > 62*1/31
nên 1/2+1/3+1/4+...+1/63>2 (dpcm)
a) \(A=\frac{7}{10}+\frac{7}{10^2}+\frac{7}{10^3}+...\)
\(A=\frac{777...}{1000...}\)
b) 1/2+1/3+1/4+…+1/63=1/2+(1/3+1/4)+(1/5+1/6+…+1/10)+(1/11+1/12+….+1/20)+(1/21+1/22+….1/63).
Ta thấy:
1/3+1/4>1/4+1/4=1/2
1/5+1/6+…+1/10>5/10=1/2
1/11+1/12+….+1/20>10/20=1/2
Thêm.cái 1/2 sắn có là đủ >2 rồi nhể
A=1+(1/2 + 1/3 + 1/4)+(1/5 + 1/6 + 1/7 + 1/8)+(1/9+...+1/16)+(1/17+...+1/32)+(1/33+...+1/64)
A>1+(1/2 + 1/4 + 1/4)+(1/8+ 1/8+ 1/8+ 1/8)+(1/16+1/16+...+1/16)+(1/64+...+1/64)
A>1 + 1 + 1/2 + 1/2 + 1/2+ 1/2
A>4
Bạn cứ xét :
\(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}>1\)
\(\Rightarrow\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+...>1\)
Vậy : \(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{63}>2\).
Cách khác đây :
Tính : \(S62=32.\left[2.\frac{1}{2}-\left(62-1\right).\left(-\frac{1}{6}\right)\right]>2\)
Vậy : \(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{63}>2\)