Clarity

And then one day, it all comes together; my long mathematical travails have
come full circle .I was keeping working on the notion of Taylor series as an interesting addendum to a lot of work on calculus. I recently stumbled upon an explanation in Math is Fun which made it clear and it is an absolutely central notion, which makes the (for me) long enigmatic Euler Formula crystal
clear. March on!

It's a beautiful thing. Every time one adds a member to the series (which is infinite), one arrives at a better approximation of the cos function. The series starts at one point and one converges to the

correct answer, which is a limit, all  notions central to Calculus.

The filiation to Euler is also quite enlightening, and clears up the confused historical record on mathematical finds in the Modern period. Quoting Wikipedia:

"The concept of a Taylor series was discovered by the Scottish mathematician James Gregory and formally introduced by the English mathematician Brook Taylor in 1715." James Gregory published

the series in 1668. But it did not originate with him either. The first publication was in the XIVth

century, in India, the work of  Madhava of Sangamagrama.

Clarity at last. The concept of an analytic function is present  at the beginning of the development of Calculus. 

Incidentally, the Kerala region of India where Madhava lived is superb, 'le pays de Dieu'.