Use the Comparison Test to determine if the series converges or diverges.
-$$\sum _ { n = 1 } ^ { \infty } \frac { 1 } { ( \ln 10 n ) ^ { n } }$$
A) converges
B) diverges

Question

Quizplus · Accepted Answer

Let $a_n=\frac{1}{(\ln 10n)^n}$. We can compare this series to the series $\sum_{n=1}^{\infty} \frac{1}{n^n}$. To see why, we can use the fact that $\ln 10 n \geq 1$, which leads to $\frac{1}{(\ln 10n)^n} \leq \frac{1}{n^n}$ for all $n \geq 1$.

We can apply the Comparison Test to $\sum_{n=1}^{\infty} \frac{1}{n^n}$ to see that it converges (perhaps by an integral test or the fact that the denominator grows so fast). Since our series is bounded by a convergent series, it converges as well. Therefore, the best choice with explanation is A.

Use the Comparison Test to Determine If the Series Converges ∑n=1∞1(ln⁡10n)n\sum _ { n = 1 } ^ { \infty } \frac { 1 } { ( \ln 10 n ) ^ { n } }n=1∑∞​(ln10n)n1​

Use the Comparison Test to Determine If the Series Converges $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { ( \ln 10 n ) ^ { n } }$