Question

E(x)=
$\frac{1}{ x+ 1} - \frac{2 + 2x}{x { }^{3} + 1} + \frac{2}{x {}^{2} - x + 1 }$
e) Calculati suma E(1)×E(2)+E(2)×E(3)+...+E(19)×E(20)

Va rog!!!!!!!​

Answer 1

Explicație pas cu pas:

$E(x)=\dfrac{1}{x + 1} - \dfrac{2 + 2x}{{x}^{3} + 1} + \dfrac{2}{{x}^{2} - x + 1} =$

$=\dfrac{1}{x + 1} - \dfrac{2(x + 1)}{{x }^{3} + 1} + \dfrac{2(x + 1)}{({x}^{2} - x + 1)(x + 1)}$

$=\dfrac{1}{x + 1} - \dfrac{2(x + 1)}{{x }^{3} + 1} + \dfrac{2(x + 1)}{{x }^{3} + 1}$

$\bf= \dfrac{1}{x + 1}$

.

E(1)×E(2)+E(2)×E(3)+...+E(19)×E(20) =

$= \dfrac{1}{2} \cdot \dfrac{1}{3} + \dfrac{1}{3} \cdot \dfrac{1}{4} + ... + + \dfrac{1}{19} \cdot \dfrac{1}{20} + \dfrac{1}{20} \cdot \dfrac{1}{21}$

$= \dfrac{1}{2} - \dfrac{1}{3} + \dfrac{1}{3} - \dfrac{1}{4} + ... + \dfrac{1}{19} - \dfrac{1}{20} + \dfrac{1}{20} - \dfrac{1}{21}$

$= \dfrac{1}{2} - \dfrac{1}{21} = \dfrac{21 - 2}{42} = \bf \dfrac{19}{42}$