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

Find the derivative and the integral of the series
.
Simplify your answer and compare with the previous example. Can you use your
results to figure out what function this power series represents?

Let
Then you can verify that:g(x) =

f(x) =

g(x) = f(x)and g(x) = -f(x) + C

There is a familiar pair of functions that have these properties. Do you remember? To help your memory, here is a plot of the functions:

f(x) =g(x) =