Question

[lu xu bu]

Chứng minh rằng

\(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+..+\frac{1}{121}-\frac{1}{122}+\frac{1}{123}=\frac{1}{62}+\frac{1}{63}+...+\frac{1}{122}\)-\(\frac{1}{123}\)

Answer 1

[Phạm Tuấn Đạt]

Xét \(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{123}\)

\(=\left(1+\frac{1}{3}+...+\frac{1}{121}+\frac{1}{123}\right)-\left(\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+...+\frac{1}{122}\right)\)

\(=\left(1+\frac{1}{3}+...+\frac{1}{121}+\frac{1}{123}\right)-2\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{61}\right)\)

\(=\frac{1}{62}+\frac{1}{63}+\frac{1}{64}+...+\frac{1}{123}\)