Question

Desenvolva.

[x-1/x]⁵​

Answer 1

$\displaystyle (\text A-\text B)^{\text n} =\sum_{\text k = 0 }^{ \text n} (-1)^{\text k}{\text n \choose \text k}\text{A}^{\text n-\text k}.\text B^{\text k } \\\\\\ \text{Fazendo n = 5} :\\\\ (\text A-\text B)^5 =\sum_{\text k=0}^{\text 5}(-1)^{\text k}{\text 5\choose\ \text k}\text A^{\text n - \text k }.\text B^{\text k} \\\\\\ (\text A-\text B)^5 = \text A^5-5\text A^4\text B+10\text A^3\text B^2-10\text A^2\text B^3+5\text A\text B^4-\text B^5$

Temos :

$\displaystyle [\ \frac{\text x-1}{\text x}\ ]^5 = \frac{(\text x-1)^5}{\text x^5} \\\\\\ \text{Desenvolvendo }: \\\\ \frac{(\text x-1)^5}{\text x^5} =\\\\\\ \huge\boxed{\frac{\text x^5-5\text x^4+10\text x^3-10\text x^2+5\text x-1}{\text x^5}}\checkmark \\\\\\ \text{ou} \\\\\\ \huge\boxed{1-\frac{5}{\text x}+\frac{10}{\text x^2}-\frac{10}{\text x^3}+\frac{5}{\text x^4}-\frac{1}{\text x^5}\ }\checkmark$