Use the Direct Comparison Test (if possible) to determine whether the series $$\sum _ { n = 9 } ^ { \infty } \frac { 1 } { n ^ { 5 / 6 } - 8 }$$
A) converges
B) diverges

Question

Quizplus · Accepted Answer

The Direct Comparison Test can be applied by comparing the given series to a simpler series that is easier to analyze. For large values of $n$, the term $n^{5/6} - 8$ behaves similarly to $n^{5/6}$, so we compare the given series to $\sum _ { n = 9 } ^ { \infty } \frac { 1 } { n ^ { 5 / 6 } }$. The series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 5 / 6 } }$ is a p-series with $p = 5/6$, which diverges because $p \leq 1$. Since the given series is larger term-by-term than a divergent series (for large $n$), it also diverges by the Direct Comparison Test.

Use the Direct Comparison Test (If Possible) to Determine Whether ∑n=9∞1n5/6−8\sum _ { n = 9 } ^ { \infty } \frac { 1 } { n ^ { 5 / 6 } - 8 }n=9∑∞​n5/6−81​

Use the Direct Comparison Test (If Possible) to Determine Whether $\sum _ { n = 9 } ^ { \infty } \frac { 1 } { n ^ { 5 / 6 } - 8 }$