= Solution
The absolute-value <series> can be grouped in pairs of nonnegative terms:
$$
\sum_{n\ge1}\bigl(|a_{2n-1}|+|a_{2n}|\bigr)
=\frac54\sum_{n\ge1}\frac1{n^2}<\infty.
$$
Indeed, $1/n^2\le\int_{n-1}^{n}x^{-2}\,dx$ for $n\ge2$, so these positive <partial sums> are bounded by two. <Completeness of the real numbers> proves their convergence. The <triangle inequality> bounds every late signed sum by the corresponding late absolute-value sum, so <absolute convergence> implies convergence. Thus \b[the <series> converges absolutely].
Back to article page
