AP® Calculus BC · Unit 10 · Topic 10.15
Differentiating and Integrating Power Series Term by Term
Inside its interval of convergence, a power series behaves like a polynomial: you can differentiate or integrate it term by term, and the result is the derivative or integral of the original function.
A power series converges to a genuine function on its interval of convergence, and on that interval the function is as well-behaved as a polynomial: smooth, with no breaks or corners. That good behavior is exactly what lets you differentiate or integrate the series one term at a time. Each term is just a power of x times a constant, so differentiating or integrating it follows the usual power rule. The new series you build this way converges to the derivative or integral of the original function, at least on the interior of the same interval.
This is how new series get built from old ones without starting from scratch. The geometric series 1/(1-x) = sum of xn for |x| < 1 is the seed. Replace x with -x and integrate to get ln(1+x). Replace x with -x2 and integrate to get arctan(x). Differentiate the plain geometric series instead, and you get a series for 1/(1-x)2, since the derivative of 1/(1-x) is 1/(1-x)2. One known series quietly generates a whole family of others.
The radius of convergence never changes under this process, but the endpoints can. Integration adds a power of x and divides by a larger number, which can tame divergence at an endpoint that the original series could not survive, so the interval can gain an endpoint. Differentiation does the reverse and can lose one. This term-by-term machinery is also the engine behind the unit's last move: once a function is written as a power series, limits and integrals that resist ordinary methods often fall apart easily, term by term, inside that series.
Key terms
- Power series
- An infinite sum of the form sum of cn (x-a)n, treated as a function of x.
- Interval of convergence
- The set of x values for which a power series converges to a finite sum.
- Radius of convergence
- Half the length of the interval of convergence, measuring how far from the center the series still converges.
- Term-by-term differentiation/integration
- Differentiating or integrating each term of a power series individually to get a new series for the derivative or integral.
Watch it explained
- How it worksHow does integrating the geometric series produce the series for ln(1 + x)?
- Worked exampleFind a power series for arctan(x) by integrating the series for 1/(1 + x2).
- Worked exampleFind the series for 1/(1 - x)2 by differentiating the geometric series.
- Common mistakeWhy isn't the interval of convergence identical after integrating term by term?
- Ties togetherHow does differentiating this power series term by term connect to using series to evaluate limits and integrals in the final topic of the unit?
Check yourself
End of topic.