Question

To determine the convergence or divergence of the series ∑k=2[infinity]k(lnk)21, use the Integral Test. Evaluate the corresponding integral and check if it converges or diverges. If the integral is finite, the series converges; if it's infinite, the series diverges.

Answer 1

The series ∑k=2[infinity]k(lnk)21 diverges.

Step-by-step explanation:

The Integral Test establishes a connection between the convergence of an infinite series and the convergence of a corresponding improper integral. For the given series ∑k=2[infinity]k(lnk)21, let's consider the corresponding integral:

∫ from 2 to infinity (x * ln(x))^2 dx

To determine convergence, we need to evaluate this integral. Utilizing integration by parts, let u = ln(x) and dv = xln(x)dx:

Applying integration by parts:

∫(x * ln(x))^2 dx = x^2 * ln(x) - ∫x(2ln(x)) dx

After further simplification, the integral becomes:

= x^2 * ln(x) - 2∫xln(x) dx

This leads us back to a similar form of the original integral. Performing integration by parts once more results in an endless loop, demonstrating that the integral diverges. Consequently, since the corresponding integral diverges, the series ∑k=2[infinity]k(lnk)21 also diverges by the Integral Test.

Therefore, based on the divergence of the corresponding integral, we conclude that the given series diverges.