= Solution
There is an indexing defect in the original PDF: the term at $n=1$ is undefined because $\ln1=0$. Thus the series as literally indexed is not defined. The natural finite-index repair is to start at $n=2$, and that repaired series still \b[diverges].
Indeed, for $n\geq2$ the terms are positive and
$$
\frac{u_{n+1}}{u_n}
=2\frac{n\ln n}{(n+1)\ln(n+1)}\longrightarrow2>1.
$$
They eventually increase at a geometric rate and do not tend to zero. Since the <terms of a convergent series tend to zero>, no choice of a finite replacement for the missing first term could make the tail converge.
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