subseries

A subseries of a given series is another series formed by selecting some of the terms of the original series in their original order. More precisely, if sum_{n=1}^\infty a_n is a series, a subseries is sum_{k=1}^\infty a_{n_k}, where n_1 < n_2 < ... are positive integers. The index set {n_k} is a subsequence of the natural numbers, and the terms included preserve the original order while omitting others.

Convergence properties of subseries depend on the nature of the original series. If the series sum a_n converges absolutely (that is, sum |a_n| converges), then every subseries also converges absolutely. Consequently, all subseries converge. Without absolute convergence, a subseries may converge or diverge; there is no general guarantee. A well-known phenomenon is that a divergent series can have convergent subseries, and a convergent series can have divergent subseries.

Examples illustrate these points. The harmonic series sum_{n=1}^\infty 1/n diverges, but choosing n_k = 2^k yields the subseries sum_{k=1}^\infty 1/2^k = 1, which converges. The alternating harmonic series sum_{n=1}^\infty (-1)^{n+1}/n converges conditionally. Its subseries consisting only of positive terms, sum_{k=1}^\infty 1/(2k-1), diverges (like the harmonic series), showing that a subseries of a conditionally convergent series may diverge.

Relation to related concepts: a subseries differs from a subsequence of the sequence of terms, though both select a subset of indices. A subseries preserves order and cannot be rearranged. See also subsequences, convergence tests, and the theory of rearrangements of series.

Subseries thus provide a tool for examining how removing terms affects convergence and for constructing convergent or divergent examples from given series.