Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2014 / Paper 1 / 19 / c

There is an indexing defect in the original PDF: the term at $n=1$ is undefined because $\ln 1=0$. 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 diverges.

Indeed, for $n\ge 2$ the terms are positive andThey 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.

Use the bound $|\sin n|\le 1$:The comparison p-series has exponent $2>1$ and converges. Hence, by the comparison test for series, the original series is absolutely convergent. No alternating-sign or monotonicity assumption about $\sin n$ is needed.