The Big Idea
A power series is an infinite polynomial. Instead of having only a finite number of terms, it can continue forever:
Because a power series behaves like a polynomial inside its interval of convergence, we can differentiate and integrate it term by term. This is one of the most useful ideas in AP Calculus BC because it lets you build new power series from ones you already know.
The key rules are surprisingly simple:
- To differentiate, differentiate every term.
- To integrate, integrate every term and include a constant of integration.
- The radius of convergence stays the same when you differentiate or integrate a power series.
- The endpoints must be checked separately, because convergence at the endpoints can change.
Differentiating a Power Series
Suppose a power series centered at is:
Differentiate each term with respect to . The constant term disappears, the next term gets multiplied by its exponent, and so on:
So the general differentiation rule is:
Notice what happened to the exponent. The original term has , and after differentiation it becomes . The exponent decreases by 1, exactly as it does with an ordinary polynomial.
Integrating a Power Series
Integration works in the opposite direction. Integrate each term separately:
The general rule is:
There is one important difference from differentiation: integration introduces a constant of integration, represented by .
What Happens to the Radius of Convergence?
This is one of the most important facts to remember for AP Calculus BC:
If a power series has radius of convergence , then its differentiated series and integrated series also have radius .
Why? Differentiating or integrating changes the coefficients of the terms, but it does not change the fundamental exponential behavior that determines the radius of convergence.
The radius stays the same, but the interval of convergence is not automatically identical. The endpoints can behave differently after differentiation or integration. Always test the endpoints of the new series separately.
Worked Examples
Differentiate the power series:
Find a power series for the integral of:
The geometric series is one of the most useful starting points for constructing new power series:
Differentiate both sides term by term.
This gives a power series representation for . The new series has the same radius of convergence, .
Use the geometric series to find a power series representation for .
Consider the power series:
This is the geometric series with radius of convergence . Its interval of convergence is because the series converges at but diverges at .
Now differentiate:
The radius is still , but the endpoints must be tested again.
Why Endpoint Testing Matters
Suppose you begin with a power series whose interval of convergence is . After differentiating or integrating, you can immediately keep the same open interval because the radius is unchanged.
But you cannot automatically carry over the endpoints.
For example, a series might converge at one endpoint before differentiation but diverge there afterward. Integration can also change endpoint behavior in the opposite direction.
So your AP Calculus BC strategy should be:
- Find the radius of convergence.
- Use the original radius for the differentiated or integrated series.
- Identify the candidate endpoints.
- Test each endpoint from scratch.
A Useful AP Calculus BC Connection
Power series are especially powerful because differentiation and integration let you create new representations from familiar ones.
For example, if you know:
then differentiating gives:
and integrating can produce additional functions. This is why memorizing a few fundamental Maclaurin series is so useful: you can manipulate them to represent many other functions.
Common Mistakes
The radius stays the same, but endpoint convergence can change. Always test the endpoints after differentiation or integration.
When finding an indefinite integral of a power series, include . If you are given an initial condition, use it to determine the constant.
When integrating , the result is . The exponent increases by 1, and you divide by that new exponent.
The derivative of a constant is zero. For example, the term contributes nothing to the differentiated series.
Term-by-term differentiation and integration are valid within the appropriate interval of convergence. Do not assume an infinite series represents its function everywhere.
Practice Problems
Differentiate the power series .
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Integrate the power series .
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The series has radius of convergence . What is the radius of convergence of its derivative?
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Differentiate and determine its interval of convergence.
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Use the known series for to find a power series for .
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