There is more to summing a series than the approach usually taught, most often at university or on preparatory courses. There are many ways to venture beyond the "classical framework." One is to find a faster way of estimating its sum.
Accelerating convergence
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One might think that a series that is not absolutely convergent—or, a fortiori, is divergent—ceases to be of any interest when seeking its possible sum. Far from it: in many cases, the summation process can accelerate convergence—in other words, turn the series into an absolutely convergent series with the same sum, thereby making it useful for approximation again. Consider, for example, the alternating harmonic series mentioned in the previous 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, roughly 10*n terms are needed to approximate ln(2) to n* significant digits. Thus, even ten significant digits require adding ten billion terms!
