Question

Chứng minh rằng

1- 1/2+ 1/3- 1/4+...+ 1/99- 1/100= 1/51+ 1/52+...+ 1/100= -1/2

Answer 1

\(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+.....+\frac{1}{99}-\frac{1}{100}\)= \(\left(1+\frac{1}{3}+....+\frac{1}{99}\right)-\left(\frac{1}{2}+\frac{1}{4}+....+\frac{1}{100}\right)\)

\(=\left(1+\frac{1}{2}+\frac{1}{3}+....+\frac{1}{99}+\frac{1}{100}\right)\)\(-2\left(\frac{1}{2}+\frac{1}{4}+....+\frac{1}{100}\right)\)

\(=\left(1+\frac{1}{2}+\frac{1}{3}+....+\frac{1}{99}+\frac{1}{100}\right)-\left(1+\frac{1}{2}+\frac{1}{3}+....+\frac{1}{50}\right)\)

\(\frac{1}{51}+\frac{1}{52}+....+\frac{1}{100}=-\frac{1}{2}\)

Answer 2

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