A series ∑aₙ converges absolutely if ∑|aₙ| converges. Absolute convergence implies convergence, but not vice versa (∑(-1)ⁿ/n converges conditionally but not absolutely). A key theorem: absolutely convergent series remain convergent after any rearrangement (to the same sum), while conditionally convergent series can be rearranged to converge to any value or diverge entirely.
Compare 1 - 1/2 + 1/3 - 1/4 + ... (converges conditionally to ln 2) with its rearrangement 1 + 1/3 - 1/2 + 1/5 + 1/7 - 1/4 + ... (converges to 3ln 2/2). Show why ∑1/n converges absolutely only after grouping.
From your study of rigorous series convergence, you know that a series ∑aₙ converges when its partial sums settle to a finite limit. But there is a deeper question: does the series converge because of genuine summability, or only because of delicate cancellation between positive and negative terms? Absolute convergence — the condition that ∑|aₙ| converges — distinguishes these two cases. When the absolute values form a convergent series, the original series is "robustly" convergent; when only the original series converges (while ∑|aₙ| diverges), the convergence is "fragile," depending critically on the arrangement of terms.
The alternating harmonic series ∑(−1)ⁿ⁺¹/n = 1 − 1/2 + 1/3 − 1/4 + ⋯ converges to ln 2 by the alternating series test. But ∑1/n = 1 + 1/2 + 1/3 + 1/4 + ⋯ is the harmonic series, which diverges. So the alternating harmonic series converges conditionally — its convergence depends entirely on the alternating signs creating enough cancellation. By contrast, ∑(−1)ⁿ/n² converges absolutely because ∑1/n² = π²/6 < ∞. Here the terms are summable on their own merits, and the signs are irrelevant to convergence.
The deepest consequence of this distinction is the Riemann rearrangement theorem: a conditionally convergent series can be rearranged to converge to any real number whatsoever, or to diverge. This is not a technical curiosity — it means the "sum" of a conditionally convergent series is an artifact of the particular ordering. The mechanism is constructive: the positive terms alone diverge to +∞ and the negative terms alone diverge to −∞ (both diverge because the absolute series diverges). To hit a target T, take positive terms until the partial sum exceeds T, then negative terms until it dips below T, and repeat. The oscillations shrink to zero because aₙ → 0, so the rearranged series converges to T. You can aim at any T, or arrange for divergence.
Absolutely convergent series are immune to this pathology. If ∑|aₙ| converges, both the positive part ∑aₙ⁺ and the negative part ∑aₙ⁻ converge independently to finite values. The sum is then the difference of two finite quantities, and rearranging terms cannot change either sub-sum. Every rearrangement converges to the same value. This is why absolute convergence is the "safe" mode of convergence: it guarantees that the sum is a genuine, order-independent quantity. In applications — Fourier series, power series, numerical computation — absolute convergence is almost always the relevant condition, because rearrangement-sensitivity would make the "sum" meaningless in any context where the order of summation might vary.
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