Consider the following series.

$$

$(1)$n$(4$n $1)3$n $+1$

n $=1$

Find the following limit$.($If the limit is infinite, enter $\text{'}$$\text{'}$ or $\text{'}-$$\text{'},$ as appropriate. If the limit does not otherwise exist, enter DNE.$)$

$$lim$$n$$$$

$4$n $13$n $+1$

$=$

Determine the convergence or divergence of the series.