Find all values of $p$ for which the series $\sum _ { n = 1 } ^ { \infty } \frac { \ln \left( n ^ { 9 } \right) } { n ^ { p } }$ converges. A) $p 1$

Question

Find all values of $p$ for which the series $\sum _ { n = 1 } ^ { \infty } \frac { \ln \left( n ^ { 9 } \right) } { n ^ { p } }$ converges. A) $p < 9$ B) $p < 1$ C) $p > 9$ D) $p > 1$

Quizplus · Accepted Answer

The series $\sum _ { n = 1 } ^ { \infty } \frac { \ln \left( n ^ { 9 } \right) } { n ^ { p } }$ can be simplified to $\sum _ { n = 1 } ^ { \infty } \frac { 9\ln(n) } { n ^ { p } }$. For convergence, we use the comparison test with the p-series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { p } }$, which converges if $p > 1$. Since $\ln(n)$ grows slower than any power of $n$, the given series converges for $p > 1$.

Find All Values Of ppp For Which the Series ∑n=1∞ln⁡(n9)np\sum _ { n = 1 } ^ { \infty } \frac { \ln \left( n ^ { 9 } \right) } { n ^ { p } }∑n=1∞​npln(n9)​

Find All Values Of $p$ For Which the Series $\sum _ { n = 1 } ^ { \infty } \frac { \ln \left( n ^ { 9 } \right) } { n ^ { p } }$