We have already noted that, by the binomial series,
Thus, with replaced by
We have previously obtained the Maclaurin expansion of :
Now, we differentiate both sides with respect to :
This result matches that found from the binomial series and demonstrates that the Maclaurin expansion of a function may be differentiated term by term to give a series which will be the Maclaurin expansion of
As we noted in Section 16.4 the derived series will have the same radius of convergence as the original series.
Find the Maclaurin expansion of and state its radius of convergence.
First write down the expansion of :
Now, by differentiation, obtain the expansion of :
Differentiate again to obtain the expansion of :
Finally state its radius of convergence:
The final series: has radius of convergence since the original series has this radius of convergence. This can also be found directly using the formula and using the fact that the coefficient of the term is .