Example 8.3.12 (a): Differentiating and Integrating Power Series
The radius of convergence turns out r = (confirm!). Thus, the series can be differentiated term by term:
So the function defined by this power series is its own derivative! We know, of course, that there is only one function with that property, namely the expontential function (can you prove that?).
Thus, we have:
f(x) = ex