Ta có 1/2 - 1/4 + 1/8 - 1/16 + 1/32 - 1/64

= ( 1/2 - 1/4 ) + ( 1/8 - 1/16 ) + ( 1/32 -  1/64)

= 1/4 + 1/16 + 1/64 

= 16 + 4 + 1 /64

= 21/64 < 21/63 = 1/3 

Vậy 1/2 - 1/4 + 1/8 - 1/16 + 1/32 - 1/64 < 1/3 ( đpcm ) Chúc bn hok tốt  . k mik nha 

Ta có :

\(\frac{1}{2}-\frac{1}{4}+\frac{1}{8}-\frac{1}{16}+\frac{1}{32}-\frac{1}{64}< \frac{1}{3}\)

\(\frac{1}{4}+\frac{1}{16}+\frac{1}{64}< \frac{1}{3}\)

\(\frac{16}{64}+\frac{4}{64}+\frac{1}{64}< \frac{1}{3}\)

\(\frac{16+4+1}{64}< \frac{1}{3}\)

\(\frac{21}{64}< \frac{1}{3}\)

=> 1/2 - 1/4 + 1/8 - 1/16 + 1/32 - 1/64 < 1/3

Đặt vế trái là A ta có

\(2A=1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\frac{1}{16}-\frac{1}{32}\)

\(3A=2A+A=1-\frac{1}{64}<1\Rightarrow A<\frac{1}{3}\)
 

Đặt A= 1/2-1/4+1/8-1/16+1/32-1/64

ta có 2A=1-1/2+1/4-1/8+1/16-1/32

2A-A=A=1-1/64=63/64

vì 63/64<1/3

=>A<1/3 (đpcm)

Ta thấy:

1/2-1/4+1/8-1/16+1/32-1/64 =

(1/2+1/4+1/8+1/16+1/32+1/64) – 2 x (1/4 + 1/16 + 1/64) =

(1 – 1/64) – 42/64 =

63/64 – 42/64 = 21/64 < 21/63=1/3

=>   1/2-1/4+1/8-1/16+1/32-1/64 < 1/3

Ta có: \(\frac{1}{2}-\frac{1}{4}=\frac{2}{4}-\frac{1}{4}=\frac{2-1}{4}=\frac{1}{4}\)

\(\frac{1}{8}-\frac{1}{16}=\frac{2}{16}-\frac{1}{16}=\frac{2-1}{16}=\frac{1}{16}\)

\(\frac{1}{32}-\frac{1}{64}=\frac{2}{64}-\frac{1}{64}=\frac{2-1}{64}=\frac{1}{64}\)

=> \(\frac{1}{2}-\frac{1}{4}+\frac{1}{8}-\frac{1}{16}+\frac{1}{32}-\frac{1}{64}\)

=\(\frac{1}{4}+\frac{1}{16}+\frac{1}{64}\)

=\(\frac{16}{64}+\frac{4}{64}+\frac{1}{64}=\frac{21}{64}\)

Ta có: \(\frac{21}{64}< \frac{21}{63}=\frac{1}{3}\)

=> \(\frac{1}{2}-\frac{1}{4}+\frac{1}{8}-\frac{1}{16}+\frac{1}{32}-\frac{1}{64}< \frac{1}{3}\)

Đặt \(A=\frac{1}{2}-\frac{1}{4}+\frac{1}{8}-\frac{1}{16}+\frac{1}{32}-\frac{1}{64}\)

\(2A=1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\frac{1}{16}-\frac{1}{32}\)

\(A+2A=1-\frac{1}{64}\)

\(3A=1-\frac{1}{64}< 1\)

=>A<1/3

=>đpcm

\(\frac{1}{2}-\frac{1}{4}+\frac{1}{8}-...-\frac{1}{64}=\frac{1}{2}-\frac{1}{2^2}+\frac{1}{2^3}-..-\frac{1}{2^6}\) = A 

2A = 1 - 1/2 + 1/2^2 - ... - 1/2^5

2A + A = 1 - 1/2 + 1/2^2 - ... - 1/2^5 + 1/2 - 1/2^2 + 1/2^3 - 1/2^4 - .. - 1/2^6

     3A  = \(1-\frac{1}{2^6}=\frac{2^6-1}{2^6}\)

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