Use the integral test to decide whether the series $\sum_{n=1}^{\infty} \frac{(\ln n)^{3}}{n}$ converges or diverges.
A)It converges
B)It diverges

Question

Quizplus · Accepted Answer

The function $ f(x) = \frac{(\ln x)^{3}}{x} $ is continuous, positive, and decreasing for $ x \geq 2 $. The integral $\int_{2}^{\infty} \frac{(\ln x)^{3}}{x} \, dx$ diverges, as it can be shown by substitution that it behaves like $\int_{2}^{\infty} \frac{1}{x} \, dx$, which diverges. Therefore, by the integral test, the series diverges.

Use the Integral Test to Decide Whether the Series ∑n=1∞(ln⁡n)3n\sum_{n=1}^{\infty} \frac{(\ln n)^{3}}{n}∑n=1∞​n(lnn)3​

Use the Integral Test to Decide Whether the Series $\sum_{n=1}^{\infty} \frac{(\ln n)^{3}}{n}$