Question

Cho A= 1/2^2 +1/3^2 + 1/4^2 + .... + 1/2015^2

Chứng minh A<3/4

Answer 1

Có : \(\dfrac{1}{2^2}\) < \(\dfrac{1}{4}\)

\(\dfrac{1}{3 ^2}\) < \(\dfrac{1}{2.3}\)

...

\(\dfrac{1}{2015^2}\) < \(\dfrac{1}{2014.2015}\)

\(\Rightarrow\) A< \(\dfrac{1}{4}\) + \(\dfrac{1}{2.3}\) + ... + \(\dfrac{1}{2014.2015}\)

= \(\dfrac{1}{4}\) + \(\dfrac{1}{2} -\dfrac{1}{3}\) + ... + \(\dfrac{1}{2014} -\dfrac{1}{2015}\)

= \(\dfrac{1}{4}+\dfrac{1}{2} -\dfrac{1}{2015}\)

=\(\dfrac{3}{4}- \dfrac{1}{2015} \)

\(\Rightarrow\)A<\(\dfrac{3}{4}\)(đpcm)

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