Determine the convergence or divergence of the series using any appropriate test from this chapter. Identify the test used. $\sum _ { n = 1 } ^ { \infty } \frac { 6 n } { n + 9 }$
A) both diverges; Ratio Test and diverges; Theorem $9.9 \left( n ^ { \text {th } } \right.$ Term Test for Divergence)
B) converges; Integral Test
C) diverges; Theorem $9.9$ ( $n ^ { \text {th } }$ Term Test for Divergence)
D) converges; $p$-series
E) diverges; Ratio Test

Question

Quizplus · Accepted Answer

The $n^{\text{th}}$ Term Test for Divergence states that if the limit of $a_n$ as $n$ approaches infinity does not equal zero, then the series $\sum a_n$ diverges. For the given series, $\lim_{n \to \infty} \frac{6n}{n+9} = 6$, which does not equal zero, indicating divergence.

Determine the Convergence or Divergence of the Series Using Any ∑n=1∞6nn+9\sum _ { n = 1 } ^ { \infty } \frac { 6 n } { n + 9 }∑n=1∞​n+96n​

Determine the Convergence or Divergence of the Series Using Any $\sum _ { n = 1 } ^ { \infty } \frac { 6 n } { n + 9 }$