Use the Integral Test to determine the convergence or divergence of the series. $$\sum _ { n = 2 } ^ { \infty } \frac { 10 } { n \sqrt { \ln n } }$$
A) diverges
B) converges
C) Integral Test inconclusive

Question

Quizplus · Accepted Answer

To use the Integral Test, we need to find a continuous, positive, and decreasing function $f(x)$ such that $f(n)=\frac{10}{n\sqrt{\ln n}}$ for all positive integers $n$. We can choose $f(x)=\frac{10}{x\sqrt{\ln x}}$.

Now, we evaluate the improper integral $\int_{2}^{\infty}f(x)dx$:
\begin{align*}
\int_{2}^{\infty}\frac{10}{x\sqrt{\ln x}}dx &= 20\int_{\ln 2}^{\infty}\frac{1}{\sqrt{u}}du\quad (u=\ln x,\ du=\frac{dx}{x}) \
&= 20\lim_{t	o\infty}\left[2\sqrt{u}ight]_{\ln 2}^{t} \
&= \infty
\end{align*}
Since the improper integral diverges, the series $\sum_{n=2}^{\infty}\frac{10}{n\sqrt{\ln n}}$ also diverges by the Integral Test. Therefore, the answer is $\boxed{	ext{(A) diverges}}$.

Use the Integral Test to Determine the Convergence or Divergence ∑n=2∞10nln⁡n\sum _ { n = 2 } ^ { \infty } \frac { 10 } { n \sqrt { \ln n } }n=2∑∞​nlnn​10​

Use the Integral Test to Determine the Convergence or Divergence $\sum _ { n = 2 } ^ { \infty } \frac { 10 } { n \sqrt { \ln n } }$