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# 12.9 Some XVIII-th Century Calculations

The following proofs that

and

use XVIII-th century standards or rigor. You should decide what parts are justified. I denote by below. By the geometric series formula,

If where , , then

so

By equating the real and imaginary parts, we get

For , this yields

Thus, , so

Hence,

Since two antiderivatives of a function differ by a constant

for some constant . When , we get

so and thus
 (12.79)

For , this gives us

(which is the Gregory-Leibniz-Madhava formula). We can rewrite (12.79) as

Again, since two antiderivatives of a function differ by a constant, there is a constant such that

For , this says

and for , this says

Subtract the second equation from the first to get

i.e.,

and thus
 (12.80)

Let Subtract (12.80) from this to get

Hence, , and then .

An argument similar to the following was given by Jacob Bernoulli in 1689 [31, p 443]. Let

Then

Subtract the second series from the first to get

Therefore,

12.81   Exercise.
a) Explain why Bernoulli's argument is not valid.
b) Give a valid argument proving that

12.82   Note. The notation was introduced by William Jones in 1706 to represent the ratio of the circumference to the diameter of a circle[15, vol2, p9]. Both Maple and Mathematica designate by Pi.

The notation was introduced by Euler in 1727 or 1728 to denote the base of natural logarithms[15, vol 2, p 13]. In Mathematica is denoted by E. In the current version of Maple there is no special name for ; it is denoted by exp(1).

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