Răspunsuri

2014-09-29T22:44:38+03:00
\dfrac{1}{1\cdot3}+\dfrac{1}{3\cdot5}+...+\dfrac{1}{2013\cdot2014}=\dfrac12\cdot\left(\dfrac11-\dfrac13+\dfrac13-\dfrac15+\dfrac15-\dfrac17+...

...+\dfrac{1}{2013}-\dfrac{1}{2015})\right)=\dfrac12\cdot\dfrac{2014}{2015}=\dfrac{1012}{2015}

Inegalitatea din enunt devine acum

\dfrac25<\dfrac{1012}{2015}<\dfrac12

care adusa la acelasi numarator devine:

\dfrac{1012}{2530}<\dfrac{1012}{2015}<\dfrac{1012}{2014} care este adevarata (dintre doua fractii cu acelasi numarator este mai mare cea cu numitorul mai mic.)