Lets consider an example. We start with a funciton, integrate it, expand it into a power series, then differentiate to get back to the original function. Every indefinite integral has a constant of integration. In particular, when we integrate the series, we can't disregard the constant of integration. It turns out to be constant term of the series.
Lets look at an example.
We start with a function , and compute its series expansion by the method described above. However, as you can see, it doesn't look llike its too good of a representation. By adding this additional term, the series becomes a airly good representation of the original function within certain bounds. Here is another example using the method described above. However, when we look at the graph of the function and its series, we see that its a good representation only for a particular interval.