How was the power series form of the exponential function disovered? Was it just observed?
By the exponential function, I mean the solution to the differential equation $\frac{df}{dx} = f$ with the intial condition $f(0) = 1$.
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2Newton's De analysi per aequationes numero terminorum infinitas of 1665 is the credited as the first statement of the series, along with the series for sine, cosine, arcsine and the logarithmic series. His proof is nonrigorous and considered rather complicated compared to that obtained using MacLaurin's formula. I am not familiar with Newton's derivation, so I cannot answer the question of how it was obtained. – nwr Jan 07 '19 at 19:03
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1@NickR Hi, Nick. De Analysi is from 1669. For the trigonometric functions Newton was the first in Europe, but the Indian Kerala school did them at least a century earlier, possibly two. Mercator did the logarithm in 1668. See What was the historical context of the development of Taylor series? – Conifold Jan 07 '19 at 21:27
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1@Conifold Gosh. It's amazing how quickly one can forget. I should have remembered Madhava and the Kerala school. I even answered a question on Madhava here on HSM not so long ago in 2016. – nwr Jan 07 '19 at 23:54