Question

Question: Use the Ratio Test or the Root Test to determine the convergence or divergence of the series.

6. [-/10 Points) DETAILS LARCALC11 9.R.065. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER Use the Ratio Test or the Root Test to determine the convergence or divergence of the series. n-1 converges diverges Submit Answer Vieno Saved Work Revert to Last Response

Answer 1

we have

$\sum_{n=1}^{\infty }\frac{3^n}{n^3}$

so,

$a_n=\frac{3^n}{n^3}$

by the ratio test,

an+1 L = lim 100 an

L= lim n+00 3n+1 (n+1) 3

$L=\lim_{n \to \infty }\left | \frac{3^1n^3}{(n+1)^3} \right |$

$L=\lim_{n \to \infty }\left | \frac{3}{(1+\frac{1}{n})^3} \right |$

$L=\left | \frac{3}{(1+\frac{1}{\infty})^3} \right |$

$L=\left | \frac{3}{(1+0)^3} \right |$

$L=3$

here L = 3 > 1,

hence the given series is diverges.