Quiz 10.3:


Question:

Determine if \( \sum \limits _{n=3} ^\infty \frac{\ln n}{n} \) converges or diverges.


Solution:

\( f(x)=\frac{\ln x}{x} \) is positive, decreasing and continuous for \( x \geq 3 \)

\( \int \limits _3 ^{\infty} \frac{\ln x}{x} dx = \lim \limits _{b \rightarrow \infty} \int \limits _3 ^b \frac{\ln x}{x} dx\)
\(u=\ln x \)
\( du=\frac{1}{x} dx \)
\( = \lim \limits _{b \rightarrow \infty} \int \limits _{\ln 3} ^{\ln b}u du =\frac{1}{2}\lim \limits _{b \rightarrow \infty} \left ( \ln ^2b -\ln ^2 3 \right ) = \infty\)

Since the integral \( \int \limits _3 ^{\infty} \frac{\ln x}{x} dx \) is divergent, the series \( \sum \limits _{n=3} ^\infty \frac{\ln n}{n} \) diverges by integral test.
JCCC
JCCC