What is so Geometric about the Geometric Series

Behold the picture:

Exhibit number 2, details on Wikipedia:

Exhibit number 3, calculation of the area under a power curve by Fermat, pilfered from Analysis by Its History, by E.Hairer and G.Wanner (Springer 1996), Sect. I.3, pp. 33-34:

From this picture it looks like the area we want to calculate is sandwiched between the sums of two geometric series:

B^{k+1}(1-R)(R^k+R^{2k+1}+\dots) \le A(B) \le B^{k+1}(1-R)(1+R^{k+1}+R^{2k+2}+\dots)

After summing the series we get

R^k B^{k+1}(1-R)/(1-R^{k+1}) \le A(B) \le  B^{k+1}(1-R)/(1-R^{k+1})

At R=1 the upper and the lower bounds come together. Unfortunately, (1-R)/(1-R^{k+1}) is undefined for R=1 since it reduces to 0/0. But fortunately Fermat knew how to make sense out of the undefined expressions like this, he knew how to differentiate, so he could figure out that A(B)=B^{k+1}/(k+1).

Had he noticed this peculiar link between differentiation, i.e., making sense of 0/0, and integration, i.e., calculating areas? I have no doubt he had, Fermat was Fermat. Had he pointed it out to his high-esteemed correspondents? I don’t know. But it was up to Isaac Barrow to expose differentiation and integration as two operations inverse to each other, and up to Newton, Leibniz and Bernoulli brothers to put his remarkable observation to work.

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2 Responses to “What is so Geometric about the Geometric Series”

  1. Alexandre Borovik Says:

    Regarding your example 3: the late Israel Gelfand loved to say, refering to Fermat’s integration of a power curve, that Fermat was the inventor of q-mathematics.

  2. misha Says:

    It sounds like a stretch, but a rather stimulating one. On the other hand, the calculation reduces to making sense of 0/0, and indicates an intimate link between differentiation and integration.

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