We can also expand about other values. The method we used above to expand a series into a geometric series works only in certain cases. If an expression does not lend itself readily to this method, there are other tricks. One is to intergrate the function, expand the anti-derivative into a series, then differentiate the result. In this indirect way we find the series.
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. We can convert the new found series into a polynomial and evaluate it to demonstrate that it take values close to original funciton. We can use the same trick in reverse too. Take a function, differentiate it, expand into a power series, and integrate. When we integrate, the constant of integration will become the constant term of the series.