Answer
Diverges
Work Step by Step
We are given $a_{n+1} = \frac{n+\ln(n)}{n+10} \times a_n$ and $a_1 = \frac{1}{2}$ and $n=1,2,3,...,n$
Find the first few elements of the sequence :
$a_{1+1} = \frac{1+\ln(1)}{1+10} \times \frac{1}{2} \Rightarrow a_2 = \frac{1}{22}$
for $a_{2+1}$ our $a_n = \frac{1}{22} \Rightarrow a_3 = \frac{2+\ln(2)}{2+10}\times\frac{1}{22} \Rightarrow a_3 = \frac{2+ln(2)}{264}$
for $a_3+1$ our $a_n = \frac{2+\ln(2)}{264} \Rightarrow $ we see that our ${a_n}$ is growing, then $a_{n+1}$ also growing
Then, $\lim\limits_{n \to \infty} \frac{n+\ln(n)}{n+10} \times a_n = \infty $ and $\ne 0$, so the series $diverges$
