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

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

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,

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

`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)`

.