1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + ...+ 1/19 - 1/20 

= ( 1 + 1/3 + 1/5 + ...+ 1/19 ) - ( 1/2 + 1/4 + ...+ 1/20 ) 

= ( 1 + 1/2 + 1/3 + 1/4 + ...+ 1/19 + 1/20 ) - 2 . ( 1/2 + 1/4 + ...+ 1/20 ) 

= ( 1 + 1/2 + 1/3 + ...+ 1/20 ) - ( 1 + 1/2 + ... + 1/10 ) 

=    1/11 + 1/12 + 1/13 + ...+ 1/20 ( Đpcm ) 
TK mk nha !!! 

\(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+.....+\frac{1}{19}-\frac{1}{20}\)

\(=\left(1+\frac{1}{3}+...+\frac{1}{19}\right)-\left(\frac{1}{2}+\frac{1}{4}+...+\frac{1}{20}\right)\)

\(=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{19}+\frac{1}{20}-2\left(\frac{1}{2}+\frac{1}{4}+...+\frac{1}{20}\right)\)

\(=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{20}-1+\frac{1}{2}+....+\frac{1}{10}\)

\(=\frac{1}{11}+\frac{1}{12}+...+\frac{1}{20}\left(đpcm\right)\)

\(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+.....+\frac{1}{19}-\frac{1}{20}\)

= \(\left(1+\frac{1}{3}+\frac{1}{5}+.........+\frac{1}{19}\right)-\left(\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+.........+\frac{1}{20}\right)\)

= \(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+......+\frac{1}{19}+\frac{1}{20}-2\left(\frac{1}{2}+\frac{1}{4}+.......+\frac{1}{20}\right)\)

= \(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+..........+\frac{1}{19}+\frac{1}{20}+1+\frac{1}{2}+.............+\frac{1}{20}\)

= \(\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+.........+\frac{1}{20}\)

Vậy biểu thức \(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+.........+\frac{1}{19}-\frac{1}{20}=\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+.......+\frac{1}{20}\)( đpcm)

\(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{19}-\frac{1}{20}\)\(=\left(1+\frac{1}{3}+\frac{1}{5}+...+\frac{1}{19}\right)\)\(-\left(\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+...+\frac{1}{20}\right)\)

\(=\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{19}+\frac{1}{20}\right)\)\(-2\left(\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+...+\frac{1}{20}\right)\)

\(=\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{20}\right)\)\(-\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{10}\right)\)

\(=\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+...+\frac{1}{20}\)

Ta có:

\(\dfrac{11}{2}.\dfrac{12}{2}.\dfrac{13}{2}.....\dfrac{20}{2}\\ =\dfrac{11.12.13.....20}{2^{10}}\\ =\dfrac{\left(11.12.13.....20\right)\left(1.2.3.....10\right)}{2^{10}\left(1.2.3.....10\right)}\\ =\dfrac{1.2.3.4.....20}{2.4.6.8.....20}\\ =\dfrac{\left(1.3.5.7.....19\right)\left(2.4.6.....20\right)}{\left(2.4.6.....20\right)}\\ =1.3.5.7.....19\)

=> Đpcm

Ta có: \(1.3.5.7....19=\frac{1}{1}.\frac{3}{1}.\frac{5}{1}.\frac{7}{1}....\frac{19}{1}\)

Mà \(1.3.5.7....19=\frac{11.12.13....20}{2.2.2....2}\)

\(\Rightarrow\frac{1}{1}.\frac{3}{1}.\frac{5}{1}.\frac{7}{1}....\frac{19}{1}=\frac{11.12.13....20}{2.2.2...2}\)

\(\Rightarrow1.3.5.7...19=\frac{11}{2}.\frac{12}{2}.\frac{13}{2}.....\frac{20}{2}\)(đpcm)

P/s: Mấy bọn ko biết giải thì câm mồm vào đừng chọn sai nha!!! (Mình không nói bạn Đức Minh Nguyễn nha)