What It Means to Represent a Function as a Power Series
A power series is an infinite polynomial. Instead of describing a function with a familiar formula such as , we can sometimes describe exactly the same function with an infinite sum of powers of .
The general form of a power series centered at is:
When we represent a function as a power series, our goal is to find the coefficients that make the infinite series equal to the original function on some interval.
The most important starting point is the geometric series. Once you know its power-series representation, you can manipulate it to create series for many other functions.
The Geometric Series: The Foundation
For , the geometric series is:
This is the key identity to remember. If you can rewrite a function so that it contains something in the form , you can often immediately turn it into a power series.
The series converges when:
For example, setting gives:
This means the function can be represented by an infinite power series centered at 0.
Method 1: Rewrite the Function as a Geometric Series
The fastest way to represent many functions as power series is to manipulate the function until it looks like .
Represent as a power series centered at 0.
Method 2: Use a Geometric Series with a Different Center
Not every power series is centered at 0. If the function contains an expression such as , we may need to build a series in powers of .
Represent as a power series centered at .
Method 3: Multiply by a Power of
Sometimes the function is almost in geometric-series form, but an extra factor of is present. In that situation, represent the simpler part as a series and then multiply every term by the extra factor.
Represent as a power series centered at 0.
Method 4: Differentiate a Known Power Series
Power series can be differentiated term-by-term within their interval of convergence. This lets us create new power-series representations from ones we already know.
Start with the geometric series and differentiate both sides:
Find a power-series representation for .
Method 5: Integrate a Known Power Series
The same idea works in the other direction. We can integrate a power series term-by-term to create a representation for another function.
Find a power-series representation for .
How to Choose the Right Method
On an AP Calculus BC problem, the hardest part is often recognizing how to create the series. Use this checklist:
- Can you rewrite the function directly as a geometric series? If yes, do that first.
- Is there an extra factor such as ? Find the simpler series and multiply term-by-term.
- Does differentiation create the function you need? Differentiate a known series.
- Does integration create the function you need? Integrate a known series.
- Is the series centered somewhere other than 0? Rewrite everything in powers of .
- Always check the interval of convergence. The algebraic manipulation does not eliminate the convergence restrictions inherited from the original series.
Why the Interval of Convergence Matters
A power-series representation is not automatically valid for every value of . The original function may exist at values where the series does not converge, and the endpoints must be checked separately.
For a geometric series, the basic restriction is:
For example, if the ratio is , then:
At the endpoints, substitute directly into the series and determine whether the resulting numerical series converges.
Common Mistakes
A series centered at must be written in powers of . Don't accidentally write powers of when the problem asks for a series centered somewhere else.
If you multiply a known series by , every term must be multiplied by . Likewise, when integrating, remember that .
A correct series formula is only part of the answer. You must know where it actually converges. Start with the convergence condition for the original geometric series and then check endpoints separately when necessary.
You can certainly find Taylor coefficients using derivatives, but many AP Calculus BC problems are much faster when you recognize a geometric-series pattern and manipulate a known series instead.
Practice Problems
Represent as a power series centered at 0. State the interval of convergence.
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Represent as a power series centered at 0.
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Use differentiation of a known power series to represent .
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Use integration of a known power series to represent .
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