Determine the convergence or divergence of the series using any appropriate test from this chapter.Identify the test used.
A)diverges;Ratio Test
B)converges;both p-series and Integral Test
C)converges;p-series
D)converges;Integral Test
E)diverges;p-series

Question

Determine the convergence or divergence of the series using any appropriate test from this chapter.Identify the test used. ​  ​
A)diverges;Ratio Test 
B)converges;both p-series and Integral Test 
C)converges;p-series 
D)converges;Integral Test 
E)diverges;p-series

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