Final answer:
Using the Integral Test, the infinite series \(\sum_{k=2}^{\infty} k(\ln(k))^p \) converges for values of \(p\) that are greater than -1.
Step-by-step explanation:
The student is asking about the convergence of an infinite series using the Integral Test. For the series \(\sum_{k=2}^{\infty} k(\ln(k))^p \), we want to find the positive values of \(p\) for which the series converges. To apply the Integral Test, the function represented by the terms of the series must be positive, continuous, and decreasing for all \(k\) greater than or equal to some number \(N\). Assuming the function meets these conditions, we can compare it to the improper integral of the corresponding continuous function \(f(x) = x(\ln(x))^p\) from \(2\) to \(\infty\).
Integrating \(f(x)\) using integration techniques, we can examine the behavior of the integral as \(x\) approaches \(\infty\). If the integral converges, so does the series. The convergence of the integral (and thus the series) depends on the value of \(p\), with the series converging for \(p > -1\) and diverging otherwise.
To estimate the value of the series, one might compute the actual value of the integral from \(2\) to some finite bound \(b\), and then use this as an approximation for the sum. However, it is important to note that the Integral Test only allows us to determine convergence or divergence, not to calculate the exact sum of the series.