Determine the convergence or divergence of the series using any appropriate test from this chapter.Identify the test used.

Question

Determine the convergence or divergence of the series using any appropriate test from this chapter.Identify the test used. ​  ​

Quizplus · Accepted Answer

The series can be written as $\sum_{n=1}^\infty \frac{1}{n^p}$ where $p= \frac{\ln(4n)}{\ln(n)}$. As $\lim_{n 	o \infty} \frac{\ln(4n)}{\ln(n)}=1$, this is a p-series with $p=1$ and hence converges. Alternatively, we can use the Integral Test and consider $$ \int_1^{\infty}\frac{1}{x^{\frac{\ln(4x)}{\ln(x)}}} dx.$$ This integral can be evaluated using substitution and it can be shown to converge. Therefore, the series converges.

Determine the Convergence or Divergence of the Series Using Any