Question

"1/2+1/3-1/4+...+1/2009-1/2010": "1/2006+1/2007+...+1/2010"

Answer 1

Sửa 1/2+1/3-1/4+...+1/2009-1/2010 thành 1-1/2+1/3-1/4+..+1/2009-1/2010;1/2006+1/2007+..+1/2010 thành 1/1006+1/1007+...+1/2010

Gọi \(A=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2009}-\frac{1}{2010}\)

\(\Rightarrow A=\left(1+\frac{1}{3}+\frac{1}{5}+...+\frac{1}{2009}\right)-\left(\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+...+\frac{1}{2010}\right)\)

\(\Rightarrow A=\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2010}\right)-\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{1005}\right)\)

\(\Rightarrow A=\frac{1}{1006}+\frac{1}{1007}+...+\frac{1}{2010}\)

Ta thấy A = 1/1006+1/2007+...+1/2010

=>A : (1/1006+1/1007+1/1008+...+1/2010) = 1