Page:Calculus Made Easy.pdf/163

 But, when $$n$$ is made indefinitely great, this simplifies down to the following:

This series is called the exponential series.

The great reason why $$\epsilon$$ is regarded of importance is that $$\epsilon^x$$ possesses a property, not possessed by any other function of $$x$$, that when you differentiate it its value remains unchanged; or, in other words, its differential coefficient is the same as itself. This can be instantly seen by differentiating it with respect to $$x$$, thus:

which is exactly the same as the original series.

Now we might have gone to work the other way, and said: Go to; let us find a function of $$x$$, such that its differential coefficient is the same as itself. Or, is there any expression, involving only powers