Power Series Expansion Of E^x

We may be familiar with the concept of differentiating a function. With the use of calculus becoming popular amongst the scientific community, mathematicians became interested in knowing if there was a function which was its own derivative. If so, then the function would be its own 2^nd derivative, 3^rd derivative and so on. This interest of mathematicians in finding such a function is not just out of curiosity. Many differential equations like

require y(x) to be such that it is its own second derivative. Leonhard Euler found that there in fact existed such a function defined as,

This is an infinite, convergent series and the base e here is an irrational number whose approximate value is 2.718282.

We must praise Euler s ingenuity here. Differentiating this function term-wise we get,

which again gives back the infinite series

This wonderful pattern of repetition occurs for 2 reasons-

(i) The careful choice of terms in the series,

(ii) The infinite number of terms in the series.

Both (i) and (ii) together allow for a mechanism by which the function self-replicates itself after every differentiation.

Now there is a point that we must give some attention to. This series is the Taylor series expansion of ex . So, wouldn t it be nice to arrive at this series from Taylor expansions? The answer is no. This is because Taylor series requires that we know the derivatives of the function at the point around which we expand the function. Note, that this is a constraint imposed on us and Leonhard Euler saves us from this mess!