Absolute Convergence, the Ratio and Root Tests, and Rearrangements

The comparison test handles nonnegative series. For series with mixed signs there are two grades of convergence. A series that converges even after every term is made positive may be reordered, multiplied, and combined like a finite sum. A series that converges only because of cancellation between positive and negative terms admits no such manipulation: reordering it can change the sum or make it diverge.

Absolute and conditional convergence

Absolute convergence is the stronger condition, and it implies ordinary convergence.

Passing to the limit in the finite triangle inequality gives its infinite version for an absolutely convergent series:

The two sums are generally different numbers; the inequality only bounds one by the other. The alternating harmonic series ${\sum }_{n=1}^{\infty }\frac{(-1{)}^n}{n}$ converges (proved below) while $\sum \frac{1}{n}$ diverges, so it is the standard example of conditional convergence.

The ratio and root tests

Both standard tests for absolute convergence compare the tail of a series to a geometric series, generalizing the constant ratio $r$ of $\sum r^n$.

Neither test gives information when its limit equals $1$. The root test uses a $\mathrm{limsup}$ and so applies to every series, making it slightly stronger than the ratio test, but the ratio test is usually easier to compute — for factorials it is immediate.

The alternating series test

Conditional convergence has its own sufficient condition. A series whose terms alternate in sign and decrease to zero converges, even when the absolute series diverges.

Conditional convergence can be arbitrarily slow: $\sum \frac{(-1{)}^n}{n^p}$ converges for every $p>0$ yet fails to converge absolutely whenever $p\le 1$.

Rearrangements

A rearrangement of $\sum x_n$ is the series $\sum x_{\sigma (k)}$ for a bijection $\sigma :\mathbb{N}\to \mathbb{N}$ — the same terms summed in a different order. Order becomes irrelevant precisely when the series converges absolutely.

Conditional convergence has no such stability: reordering a conditionally convergent series can produce any sum.

The mechanism is that the positive terms alone sum to $+\infty$ and the negative terms alone to $-\infty$; only their interleaving produces a finite sum. For a target $L$, add positive terms in order until the running sum first exceeds $L$, then add negative terms until it drops below $L$, and repeat. Because the terms tend to zero, each overshoot shrinks, and the running partial sum is eventually trapped within any tolerance of $L$.⁶

Multiplying series: the Cauchy product

Multiplying two series is not term-by-term; the correct product collects all cross terms of a given total index.

The absolute-convergence hypothesis cannot be dropped.

Summary

Absolute convergence is the well-behaved case: the ratio and root tests detect it, and rearrangement and the Cauchy product preserve the sum. Conditional convergence, detected by the alternating series test, depends on the exact order of the terms; by Riemann's theorem, reordering can produce any sum.