Use the integral test to determine whether the series converges.
-$$\sum _ { n = 1 } ^ { \infty } \frac { 1 } { ( \ln 5 ) ^ { n } }$$
A) diverges
B) converges

Question

Quizplus · Accepted Answer

We can use the integral test to determine convergence or divergence of the series. Let
$$f(x) = \frac{1}{(\ln 5)^x}$$
Then,
$$\int_1^\infty f(x) dx = \int_1^\infty \frac{1}{(\ln 5)^x} dx = \lim_{n \to \infty} \frac{1}{\ln 5}\frac{1}{n-1}[(\ln 5)^{n-1}-1]$$
As $n$ approaches infinity, the expression above approaches $\infty$.

Since $\int_1^\infty f(x) dx = \infty$, the series $\sum_{n=1}^\infty \frac{1}{(\ln 5)^n}$ diverges by the integral test. Therefore, the correct answer is B.

Use the Integral Test to Determine Whether the Series Converges ∑n=1∞1(ln⁡5)n\sum _ { n = 1 } ^ { \infty } \frac { 1 } { ( \ln 5 ) ^ { n } }n=1∑∞​(ln5)n1​

Use the Integral Test to Determine Whether the Series Converges $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { ( \ln 5 ) ^ { n } }$