We integrate power series term by term.
Integration of Power Series
Suppose that the power series converges for all in some open interval . Then, on this interval, the power series has an anti-derivative which can be obtained by integrating term-by-term:
In other words, the indefinite integral of a power series is computed term by term, as we would anti-differentiate a polynomial.
example 1 The power series converges on the interval . On that interval, its
anti-derivative is given by The interval of convergence of the anti-derivative is . In
general, the anti-derivative might converge at an endpoint when the original series
did not.
(problem 1) Find the anti-derivatives of the following power series. Also,
compare the interval of convergence of the original series to its anti-derivative.
example 2 The power series expansion for the function is Integrate the
function and the series to obtain a power series representation for a logaritmic
function.
First, note that Integrating the power series: Equating these two antiderivatives, we have If we set , we can find : so . Hence, and we have a power series representiation for a logarithmic function.
First, note that Integrating the power series: Equating these two antiderivatives, we have If we set , we can find : so . Hence, and we have a power series representiation for a logarithmic function.
example 3 The power series representation for the function is Integrate the function
and the series to obtain a power series representation for an inverse trigonometric
function.
We have and integrating the power series gives: Equating these two antiderivatives, we have If we set , we can find : so and we have the following power series representation for the inverse tangent function:
We have and integrating the power series gives: Equating these two antiderivatives, we have If we set , we can find : so and we have the following power series representation for the inverse tangent function: