Euler's formula for ex
We want to prove the identity
If we consider the Taylor series for
, on the left hand side we have
On the right, we can expand out the product using the Binomial Theorem
The coefficient of
each term here containing
or exactly the Taylor series for
You might think this argument has made somewhat cavalier use of limits. The following argument, adapted from [1, pg 272], is close to Euler's original. He was even more bold in his use of limits!
"Euler unhesitatingly accepts the existence of both infinitely small and infinitely large numbers, and uses them to such effect that the modern reader's own hesitation must be tinged with envy." [1, pg 272]
Thinking about near , and its gradient, we see that, for infinitely small
This is illustrated above. Let
be any given number, then
is infinitely large. So
 C.H. Edwards, The Historical Development of the Calculus, Springer-Verlag, 1979.
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