Solution
= Solution
For a <series> $\sum u_k$ with eventually nonzero terms, suppose
$$
L=\lim_{k\to\infty}\left|\frac{u_{k+1}}{u_k}\right|
$$
exists, allowing $L=+\infty$. The <ratio test> says that \b[if $L<1$, the <series> converges absolutely; if $L>1$, it diverges; if $L=1$, the test is inconclusive]. In the divergent case the terms fail to tend to zero. For example, the <P-series> with exponents $1$ and $2$ both have ratio limit $1$ but have different convergence behaviour.