= Solution
The <integral test for convergence> applies when $a_n=h(n)$ for all sufficiently large $n$, where $h$ is nonnegative, continuous and nonincreasing on $[N,\infty)$. It states that $\sum_{n=N}^\infty a_n$ converges if and only if the <improper integral> $\int_N^\infty h(x)\,dx$ is finite. Its comparison bounds are
$$
\int_N^{M+1}h(x)\,dx\le\sum_{n=N}^M h(n)\le h(N)+\int_N^M h(x)\,dx.
$$
No conclusion about an arbitrary nonnegative sequence follows without such a monotone comparison function; finite initial terms do not affect convergence.
For $h(x)=x^{-\alpha}$ with $\alpha>0$, all the hypotheses hold. If $\alpha\ne1$,
$$
\int_1^R x^{-\alpha}\,dx=\frac{R^{1-\alpha}-1}{1-\alpha},
$$
whereas for $\alpha=1$ the integral is $\log R$. The first expression has a finite limit exactly when $\alpha>1$. Thus the <p-series> satisfies
$$
\boxed{\sum_{n=1}^\infty n^{-\alpha}\text{ converges iff }\alpha>1;\quad\text{it diverges for }0<\alpha\le1.}
$$
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