Question

Cho A=\(\frac{1}{1.2.3}\) + \(\frac{1}{2.3.4}\) + \(\frac{1}{3.4.5}\) + ... + \(\frac{1}{2014.2015.2016}\). So sánh A với \(\frac{1}{4}\).

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Answer 1

2A=\(\frac{2}{1\cdot2\cdot3}\)+\(\frac{2}{2\cdot3\cdot4}\)+\(\frac{2}{3\cdot4\cdot5}\)+...+\(\frac{2}{2014\cdot2015\cdot2016}\)

2A=\(\frac{1}{1\cdot2}\)-\(\frac{1}{2\cdot3}\)+\(\frac{1}{2\cdot3}\)-\(\frac{1}{3\cdot4}\)+\(\frac{1}{3\cdot4}\)-\(\frac{1}{4\cdot5}\)+...+\(\frac{1}{2014\cdot2015}\)-\(\frac{1}{2015\cdot2016}\)

2A=\(\frac{1}{2}\)-\(\frac{1}{2015\cdot2016}\)

A=(\(\frac{1}{2}\)-\(\frac{1}{2015\cdot2016}\)):2

A=\(\frac{1}{2}\):2-\(\frac{1}{2015\cdot2016}\):2

A=\(\frac{1}{4}\)-\(\frac{1}{2015\cdot2016\cdot2}\)<\(\frac{1}{4}\)

Vậy A<\(\frac{1}{4}\)