Power series let us write many functions as infinite polynomials, which are often easier to approximate, differentiate, integrate, and analyze. Building new series means starting from a familiar series and transforming it into a series for a new function. This matters because one known pattern, such as the geometric series, can generate many useful expansions.
The main tools are substitution, multiplication, term-by-term differentiation, and term-by-term integration.
The most important starting point is 1/(1 - x) = sum from n = 0 to infinity of x^n, valid for |x| < 1. By replacing x with another expression, multiplying by powers or constants, or differentiating and integrating each term, you can create new power series with predictable coefficients. The radius and interval of convergence must be checked after every transformation, especially at endpoints.
A worked example often follows the path from a known input series through an operation, then to a simplified output series.
Key Facts
- Geometric series: 1/(1 - x) = sum from n = 0 to infinity of x^n, valid for |x| < 1.
- Substitution rule: replace x by g(x) to get 1/(1 - g(x)) = sum from n = 0 to infinity of (g(x))^n, valid when |g(x)| < 1.
- Term-by-term differentiation: if f(x) = sum a_n x^n, then f'(x) = sum from n = 1 to infinity of n a_n x^(n - 1) inside the radius of convergence.
- Term-by-term integration: if f(x) = sum a_n x^n, then integral f(x) dx = C + sum from n = 0 to infinity of a_n x^(n + 1)/(n + 1) inside the radius of convergence.
- Multiplying by x^k shifts powers: x^k sum a_n x^n = sum a_n x^(n + k).
- Endpoint convergence must be tested separately because differentiation and integration keep the same radius of convergence but may change endpoint behavior.
Vocabulary
- Power series
- A power series is an infinite sum of the form sum a_n (x - c)^n, where c is the center and a_n are coefficients.
- Geometric series
- A geometric series is a series with a constant ratio between consecutive terms, such as sum x^n = 1/(1 - x) for |x| < 1.
- Radius of convergence
- The radius of convergence is the distance from the center within which a power series is guaranteed to converge.
- Term-by-term differentiation
- Term-by-term differentiation means differentiating each term of a power series to create a series for the derivative.
- Interval of convergence
- The interval of convergence is the set of x-values where a power series converges, including any endpoints that pass separate tests.
Common Mistakes to Avoid
- Forgetting to update the convergence condition after substitution is wrong because |x| < 1 becomes a new condition such as |2x| < 1 or |x^2| < 1.
- Differentiating the function but not the series terms correctly is wrong because each term a_n x^n becomes n a_n x^(n - 1), and the n = 0 term disappears.
- Assuming endpoints are automatically included is wrong because the radius gives an open interval first, and each endpoint must be tested with the resulting numerical series.
- Losing the constant of integration is wrong because an antiderivative series should include C unless a specific initial value determines it.
Practice Questions
- 1 Use 1/(1 - x) = sum from n = 0 to infinity of x^n to find a power series for 1/(1 + 3x). State the interval of convergence.
- 2 Starting with 1/(1 - x), differentiate term by term to find a power series for 1/(1 - x)^2. Then write the first four nonzero terms.
- 3 A student replaces x with x^2 in the geometric series and says the interval of convergence is still -1 < x < 1 because the original series used |x| < 1. Explain the correct reasoning and whether the final interval changes.