Representing Functions as Power Series

Consider

                                     (1)

EXAMPLE 1: Express as a power series and find the interval of convergence.


EXAMPLE 2: Express as a power series and find the interval of convergence.


EXAMPLE 3: Express as a power series and find the interval of convergence.


EXAMPLE 4: Express as a power series and find the interval of convergence.


EXAMPLE 5: Express as a power series and find the interval of convergence.



THEOREM (Differentiation and Integration of Power Series): If the power series



has radius of convergence then the function defined by



is differentiable (and therefore continuous) on an interval and



The radii of convergence of the power series in Equations (i) and (ii) are both

REMARK: Although the Theorem above says that the radius of convergence remains the same when a power series is differentiated or integrated, this does not mean that the interval of convergence remains the same. It may happen that the original series converges at an endpoint, whereas the differentiated series diverges there.

EXAMPLE 6: Express as a power series and find its radius of convergence.


EXAMPLE 7: Express as a power series and find its radius of convergence.


REMARK: One can show that formula (2) from the example above is also true if Substituting into the formula, we get



EXAMPLE 8: Express as a power series and find its radius of convergence.


REMARK: One can show that formula (3) from the example above is also true if Substituting into the formula, we get



EXAMPLE 9: Express as a power series.