Determine if the series converges absolutely, converges, or diverges.
-$$\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { 9 n ^ { 4 / 3 } + 8 }$$
A) Converges conditionally
B) Converges absolutely
C) diverges

Question

Quizplus · Accepted Answer

The series converges absolutely because the terms $\frac{1}{9n^{4/3} + 8}$ decrease and approach zero as $n$ approaches infinity, and the series $\sum \frac{1}{n^{4/3}}$ is a p-series with $p > 1$, which converges. Therefore, the original series also converges absolutely, regardless of the alternating sign.

Determine If the Series Converges Absolutely, Converges, or Diverges ∑n=1∞(−1)n9n4/3+8\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { 9 n ^ { 4 / 3 } + 8 }n=1∑∞​9n4/3+8(−1)n​

Determine If the Series Converges Absolutely, Converges, or Diverges $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { 9 n ^ { 4 / 3 } + 8 }$