Our understanding of the sum of a series need not be limited to the way it is usually introduced, most often at university or in preparatory classes. There are many ways to move beyond the “classical framework”. One is to find a way to obtain an estimate of the sum more quickly.
Speeding up convergence
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One might think that a series that is not absolutely convergent (or, a fortiori, divergent) is of no use in approximating its possible sum. This is not so: in many cases, a summation method can accelerate convergence; in other words, it can transform the series into an absolutely convergent series with the same sum and restore its approximation property. Consider, for example, the alternating harmonic series, mentioned in the preceding articles, whose sum in the classical sense is the natural logarithm of 2:
1 – 1/2 + 1/3 – 1/4 + 1/5 – 1/6 … = ln(2) = 0,69314…
Convergence is slow. In other words, many terms, of the order of 10*n, are needed to obtain an approximation of ln(2) to n* significant figures. Thus, ten billion terms must be added here to obtain only ten significant figures!
