Converging to a number: is there only one way?

To study a numerical series, we begin with a sequence (*u_n)_n*_≥0 of numbers (real or complex). The sequence converges to a limit L if, for every ε, however small, there exists an index N beyond which the difference |*u_n – L| is less than ε.*

From a sequence to a convergent series --------------------------------------

The same principle applies to the convergence of a series. We define the sequence of partial sums ${\text{S}}_n={\sum }_{k=0}^n{u}_k\backslash text\{S\}\_n\; =\; \backslash sum\_\{k=0\}^n\; u\_k$ and see whether it converges to a limit S, which, if it exists, is precisely the sum of the series.

A necessary condition for the series Σ *u_n to converge is that its general term u_n tend to 0, but this condition is not sufficient. The classic example is the harmonic series Σ 1/n*, which diverges because its partial sums tend to infinity.

A series of positive terms may diverge, however, while a series with the same terms but alternating signs (+ and –) converges. This is true, for example, of alternating series (the + and – signs alternate, and the absolute values of the terms form a decreasing sequence). For such sequences, when the general term *u_n tends to 0, even if the series Σ u_n diverges, the alternating series Σ (‒1)ⁿu_n converges. The best-known example is the alternating harmonic series Σ (‒1)ⁿ/n*, which features prominently in this special issue.