What Is a Power Series?
A power series is an infinite series whose terms contain powers of a variable. The simplest example is:
This looks like the familiar geometric series, except that the ratio between consecutive terms contains the variable . Because the value of can change, the series may converge for some values of and diverge for others.
More generally, a power series centered at has the form:
The number is called the center of the power series. The expression is the power of the variable, and is the coefficient of that power.
A power series is essentially a polynomial with infinitely many terms. Instead of stopping after a finite number of powers, we keep adding powers of indefinitely.
Writing a Power Series in Sigma Notation
Writing every term of an infinite series would be impractical. Sigma notation gives us a compact way to describe the entire pattern.
For example, consider:
The pattern is clear: the exponent increases by 1 each time. We can write the same series as:
Here, tells us that the exponent of is always the current value of .
Recognizing the Pattern
- Look at the coefficient of each term.
- Look at the exponent of the variable.
- Determine how both patterns change as the index increases.
- Write the general term using .
The Geometric Series Connection
The most important power series to recognize initially is the geometric series:
This converges when the absolute value of the common ratio is less than 1:
If the ratio is , then the power series becomes:
Therefore, it converges when:
This gives us one of the most important power series in AP Calculus:
Inside its interval of convergence, this infinite series is actually equal to the function:
Power series let us represent functions as infinite polynomials. Once we know the series, we can often differentiate or integrate it term-by-term and use it to approximate the original function.
Radius of Convergence
A power series does not usually converge for every value of . Instead, there is often a region around the center where it converges.
For a power series centered at , the radius of convergence is the number such that the series converges for:
and diverges outside that interval:
The endpoints and must always be checked separately.
Inside the radius: The series converges.
Outside the radius: The series diverges.
At the endpoints: You must test each endpoint individually.
Finding the Radius with the Ratio Test
The ratio test is one of the most useful tools for finding the radius of convergence.
Suppose we have:
Apply the ratio test by examining:
For convergence, this limit must be less than 1. Solving that inequality gives the interval where the series converges.
Worked Examples
Consider the power series:
Find the radius of convergence of:
Find the interval of convergence for:
What function is represented by:
Power Series as Infinite Polynomials
It is useful to think of a power series as an infinite polynomial:
A finite polynomial has a finite number of powers of . A power series continues indefinitely. This gives us a powerful connection between polynomial algebra and calculus.
Within its interval of convergence, a power series can be differentiated term-by-term:
It can also be integrated term-by-term:
These ideas eventually lead directly to Taylor series and Maclaurin series, which are among the most important topics in AP Calculus BC.
Common Mistakes
Finding the radius only gives the behavior inside and outside the radius. The endpoints can behave differently from each other, so substitute each endpoint into the original series and test it.
The radius is a single positive number. The interval of convergence is the actual set of -values for which the series converges. For example, a radius of 3 centered at 2 gives the initial interval , before endpoint testing.
The ratio test often gives an open interval. That is not automatically the final answer. Always go back and test both endpoints.
Some power series have coefficients that vary with , such as factorials or binomial coefficients. In those cases, the ratio test or another convergence test is usually needed.
A Reliable AP Calculus BC Strategy
- Identify the power-series form. Look for powers of or .
- Find the center. Rewrite the expression as powers of whenever possible.
- Find the radius. Use the geometric-series condition or the ratio test.
- Write the candidate interval.
- Test both endpoints. Never assume they behave the same way.
- Recognize familiar series. Geometric series are especially important because they can often be converted directly into functions.
Power series → Center → Radius → Candidate interval → Endpoint tests → Final interval of convergence
Practice Problems
Find the radius of convergence of .
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Find the interval of convergence of .
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Determine the function represented by .
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Find the interval of convergence of .
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