Question

sobre a série E com n=1 , 5^n / n² , pode-se afirmar que

Answer 1

$\rho= \lim_{n\to \infty } \frac{a_{n+1}}{a_n} \\\\ \bmatrix \rho\ \textless \ 1 \to \text{converge} \\ \rho \ \textgreater \ 1 \to \tex{diverge}\\ \rho=1 \to\text{inconclusivo} \end$


$\rho= \lim_{n\to \infty } \frac{\frac{5^{n+1}}{(n+1)^2}}{\frac{5^n}{n^2}} \\\\ \rho=\lim_{n\to \infty } \frac{5^{n+1}}{(n+1)^2} * \frac{n^2}{5^n}\\\\ \rho= \lim_{n\to \infty } \frac{5n^2}{(n+1)^2} \\\\ \rho=\lim_{n\to \infty } \frac{5n^2}{(n+1)^2} \\\\\rho= lim_{n\to \infty } \;5*(\frac{n}{(n+1)})^2 \\\\ \rho= lim_{n\to \infty } \;5*(\frac{1}{(1+ \frac{1}{n} )})^2 \\\\ \rho = 5* (\frac{1}{1+0})^2 = 5 \to \text{diverge}$