Question

Use the Integral Test to determine the convergence or divergence of the series. $$ \sum_{n=2}^{\infty} \frac{\ln n}{n^{3}} $$

Short answer

The series \( \sum_{n=2}^{\infty} \frac{\ln n}{n^{3}} \) is convergent.

Step-by-step solution

Rewrite the series as a function

The series can be rewritten as a function \( f(x) = \frac{\ln x}{x^{3}} \).

Apply the Integral Test

To apply the Integral Test, integrate \( f(x) \) from 2 to infinity:\n \[ \int_{2}^{\infty} \frac{\ln x}{x^{3}} dx \].

Evaluate the integral

To evaluate this integral, we can use integration by parts. Let \( u = \ln x, dv = \frac{dx}{x^{3}} \). Then \( du = \frac{dx}{x}, v = -\frac{1}{2x^{2}} \).\n After substituting these into the integral, we have:\n \[ -\frac{1}{2} \Bigg[\frac{\ln x}{x^{2}}\Bigg]_{2}^{\infty} + \frac{1}{2} \int_{2}^{\infty} \frac{1}{x^{2}} dx \] \n Evaluate both terms separately. The first term tends towards 0 as \( x \) approaches infinity and is finite for \( x = 2 \). The second term is a proper integral and its value is finite. Therefore, the original integral is finite.

Determine the convergence

Since the integral of \( f(x) \) from 2 to infinity is finite, the Integral Test tells us that the original series \( \sum_{n=2}^{\infty} \frac{\ln n}{n^{3}} \) is convergent.