Question

imi poate arata cineva cum se rezolva,vreau sa inteleg cum sa fac?Multumesc.​

Answer 1

Explicație pas cu pas:

$\boxed {\dfrac{1}{ {n}^{2} } < \dfrac{1}{n(n - 1)}} \ \ \boxed {\dfrac{1}{n(n - 1)} = \dfrac{1}{n - 1} - \dfrac{1}{n}}$

$\dfrac{1}{4} + \dfrac{1}{9} + \dfrac{1}{16} + ... + \dfrac{1}{ {n}^{2} } =$

$= \dfrac{1}{ {2}^{2} } + \dfrac{1}{ {3}^{2} } + \dfrac{1}{ {4}^{2} } + ... + \dfrac{1}{ {n}^{2} } < \dfrac{1}{2 \cdot 1} + \dfrac{1}{3 \cdot 2} + \dfrac{1}{4 \cdot 3} + ... + \dfrac{1}{ {n}^{2} }$

$= \Big(\dfrac{1}{1} - \dfrac{1}{2}\Big) + \Big(\dfrac{1}{2} - \dfrac{1}{3}\Big) + \Big(\dfrac{1}{3} - \dfrac{1}{4}\Big) + ... + \Big(\dfrac{1}{n - 1} - \dfrac{1}{n}\Big)$

$= \dfrac{1}{1} - \dfrac{1}{n} = \dfrac{n - 1}{n} \leqslant 1$

q.e.d.