Ta cóC= \(\dfrac{1}{2}.\dfrac{3}{4}.\dfrac{5}{6}......\dfrac{9999}{10000}\)

Đặt A = \(\dfrac{2}{3}.\dfrac{4}{5}.\dfrac{6}{7}.....\dfrac{10000}{10001}\)

Khi đó AC = \(\dfrac{1}{2}.\dfrac{2}{3}.\dfrac{3}{4}.....\dfrac{9999}{10000}.\dfrac{10000}{10001}\)= \(\dfrac{1}{10001}\)

Do \(\dfrac{1}{2}< \dfrac{2}{3}\)

\(\dfrac{3}{4}< \dfrac{4}{5}\)

.............

\(\dfrac{9999}{10000}< \dfrac{10000}{10001}\)

=> C<A=>C²<CA hay C²< \(\dfrac{1}{10001}\) , mà \(\dfrac{1}{10001}\)<\(\dfrac{1}{10000}\)=> C²< \(\dfrac{1}{10000}\)

^{Khi đó C < \(\sqrt{\dfrac{1}{10000}}\)}hay C< \(\dfrac{1}{100}\)( đpcm )

 Đặt A = (1/2)(3/4)(5/6) ... (9999/10000) (A > 0) 
.Và B = (2/3)(4/5)(6/7) ... (10000/10001) (B > 0) 
Ta có A.B = (1/2)(2/3)(3/4) ... (10000/10001) = 1/10001 (1) 
Mặt khác : 
1/2 < 2/3 
3/4 < 4/5 
................ 
................ 
9999/10000 < 10000/10001 
Nhân tất cả vế theo vế ---> A < B ---> A² < A.B (2) 
(1),(2) ---> A² < 1/10001 ---> A < căn(1/10001) < căn(1/10000) = 1/100 (đpcm)

Ta có C = \(\frac{1}{2}.\frac{3}{4}.\frac{5}{6}...\frac{9999}{10000}\)

Gọi D = \(\frac{2}{3}.\frac{4}{5}.\frac{6}{7}...\frac{10000}{10001}\)

Mà \(\frac{1}{2}< \frac{2}{3};\frac{3}{4}< \frac{4}{5};\frac{5}{6}< \frac{6}{7};...;\frac{9999}{10000}< \frac{10000}{10001}\)

=> C<D

Lại có C.D = \(\left(\frac{1}{2}.\frac{3}{4}.\frac{5}{6}...\frac{9999}{10000}\right)\left(\frac{2}{3}.\frac{4}{5}.\frac{6}{7}...\frac{10000}{10001}\right)\)

C.D = \(\frac{1}{2}.\frac{2}{3}.\frac{3}{4}.\frac{4}{5}.\frac{5}{6}.\frac{6}{7}...\frac{9999}{10000}.\frac{10000}{10001}\)

C.D = \(\frac{1}{10001}\)

Vì C<D

=> C.C < C.D

hay  C.C <\(\frac{1}{10001}\)

=> C < \(\frac{1}{10001}< \frac{1}{100}\)(đpcm)

1/2.3/4.5/6....9999/10000

=1/2

ta có : 

1/2 < 2/3

2/3 <3/4

.........

9999/10000 < 10000/10001

suy ra : A² < 1/2*2/3*3/4******9999/10000*10000/10001

suy ra : A² < 1/10001 < 1/10000= (1/100)²

suy ra A² < (1/100)² . Từ đó:  A < 1/100 = 0,01

bài toán ddc chứng minh!!!!!!!!!!! ****

tại sao A^2<1/10000 mà không phải là 1/10001

Vì 1/10001<1/10000

ta có : 

1/2 < 2/3

2/3 <3/4

.........

9999/10000 < 10000/10001

suy ra : A² < 1/2*2/3*3/4******9999/10000*10000/10001

suy ra : A² < 1/10001 < 1/10000= (1/100)²

suy ra A² < (1/100)² . Từ đó:  A < 1/100 = 0,01