Euler was the first to find the exponential series which came out of examining compound interest.
2.
He is known for having calculated the solution to a chessboard problem involving an exponential series.
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To differentiate the associated exponential series, the sequence of coefficients needs to be shifted one place to the " left " ( losing the first term ).
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So in a certain sense the exponential series is an instance of the geometric series .-- a 08 : 39, 14 January 2012 ( UTC)
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Here \ scriptstyle T ^ k [ 1 ] is the k-th iterated integral of the constant 1, that is \ scriptstyle x ^ k / k !, and we get the exponential series.
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It might appear that this is the non-constructive equivalent of the constructive definition using iterated exponentiation; but the two definitions are equally non-constructive at steps indexed by limit ordinals, which represent transfinite recursion of a higher order than taking the supremum of an exponential series.
7.
:( @ OP ) In fact, with a very small effort of abstraction, you can define exp ( z ) independently from the trigonometric functions ( e . g . exp ( z ) : = lim n?! " ( 1 + z / n ) n or by the exponential series ), then prove all its elementary properties, and " then " define cos ( x ) and sin ( x ) as the real and the imaginary part of exp ( ix ), that is the true definition.
Meaning
a series derived from the expansion of an exponential expression
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