## Example 8.3.12 (a): Differentiating and Integrating Power Series

Find the derivative of the series

*f(x) =*. What is the radius of convergence? Make sure to simplify your answer, which should be surprising. Can you use your result to figure out what function this power series represents?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:

e^{x}=

f(x) = e^{x}