A power series is an infinite polynomial used to represent a function near a chosen center point. It has the form f(x) = Σ c_n(x - a)^n, where a is the center and the numbers c_n control the size of each term. Power series matter because they let us approximate complicated functions with polynomials, which are easier to calculate, graph, and analyze.
They are a major tool in calculus, physics, engineering, and numerical computation.
Each term in a power series adds another layer of detail to the function, like adding more parts to a machine that improves its output. The center a is the point around which the series is built, and the farther x moves from a, the more important convergence becomes. The coefficients c_n often come from derivatives of the function at the center, especially in Taylor series.
When the series converges, its infinite sum defines a function on an interval around the center.
Key Facts
- General power series: f(x) = Σ c_n(x - a)^n, from n = 0 to infinity.
- The center is a, the value of x where the powers (x - a)^n are built.
- The constant term is c_0 because (x - a)^0 = 1.
- A Taylor series has coefficients c_n = f^(n)(a) / n!.
- A power series converges for |x - a| < R, where R is the radius of convergence.
- Inside its interval of convergence, a power series can be differentiated and integrated term by term.
Vocabulary
- Power series
- An infinite sum of terms in the form c_n(x - a)^n that can represent a function near the center a.
- Center
- The value a in a power series that marks the point around which the polynomial terms are built.
- Coefficient
- A number c_n that multiplies a power of (x - a) and controls that term's contribution to the series.
- Radius of convergence
- The distance R from the center within which the power series is guaranteed to converge.
- Taylor series
- A power series whose coefficients are determined by the derivatives of a function at the center.
Common Mistakes to Avoid
- Forgetting the center a, which is wrong because powers are based on (x - a), not always on x.
- Assuming every power series works for all x, which is wrong because most power series only converge inside a limited interval.
- Treating c_n as powers of x, which is wrong because c_n are coefficients that multiply the powers of (x - a).
- Ignoring endpoint checks, which is wrong because the radius test usually does not decide whether the series converges at x = a + R or x = a - R.
Practice Questions
- 1 For the power series Σ 3^n(x - 2)^n from n = 0 to infinity, find the center and radius of convergence.
- 2 Find the first four nonzero terms of the Taylor series for e^x centered at a = 0.
- 3 Explain why adding more terms of a convergent power series usually improves the approximation near the center but may not help outside the interval of convergence.