= Solution
For each $k\geq0$, monotonicity gives the dyadic-block bounds
$$
2^k x_{2^{k+1}}
\leq\sum_{n=2^k}^{2^{k+1}-1}x_n
\leq2^k x_{2^k}.
$$
Summing the upper bounds proves convergence of $\sum x_n$ whenever $\sum2^kx_{2^k}$ converges. Summing the lower bounds gives, up to the first term,
$$
\sum_{k\geq0}2^kx_{2^{k+1}}
=\frac12\sum_{j\geq1}2^jx_{2^j},
$$
so convergence of $\sum x_n$ forces convergence of the condensed series. This is the <Cauchy condensation test>:
$$
\boxed{\sum_{n\geq1}x_n\text{ converges}
\iff\sum_{k\geq0}2^kx_{2^k}\text{ converges}.}
$$
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