Power series let us write many functions as infinite polynomials, which makes calculus easier to do and easier to approximate. Term-by-term differentiation and integration mean taking the derivative or antiderivative of each separate power term in the series. This matters because it turns difficult functions into manageable algebraic patterns.
It is a core tool for Taylor series, approximations, differential equations, and physics models near an expansion point.
If a power series converges for |x - a| < R, then its derivative series and integral series have the same radius of convergence R. Inside that interval, the operations are safe because power series converge uniformly on smaller closed intervals. Endpoints must be checked separately because differentiating or integrating can change convergence there.
For example, from 1/(1 - x) = sum from n = 0 to infinity of x^n, we can integrate term by term to get -ln(1 - x) = sum from n = 0 to infinity of x^(n + 1)/(n + 1) for |x| < 1.
Key Facts
- If f(x) = sum from n = 0 to infinity of c_n(x - a)^n, then f'(x) = sum from n = 1 to infinity of n c_n(x - a)^(n - 1).
- An antiderivative is integral f(x) dx = C + sum from n = 0 to infinity of c_n(x - a)^(n + 1)/(n + 1).
- The original series, derivative series, and integral series all have the same radius of convergence R.
- The interval endpoints x = a - R and x = a + R must be tested separately after differentiating or integrating.
- Geometric series formula: 1/(1 - x) = sum from n = 0 to infinity of x^n for |x| < 1.
- Worked example: If f(x) = sum from n = 0 to infinity of x^n, then f'(x) = sum from n = 1 to infinity of n x^(n - 1) = 1/(1 - x)^2 for |x| < 1.
Vocabulary
- Power series
- A series of the form sum c_n(x - a)^n, where a is the center and c_n are constants.
- Radius of convergence
- The nonnegative number R such that a power series converges for |x - a| < R and diverges for |x - a| > R.
- Interval of convergence
- The set of x-values where a power series converges, including any endpoints that pass separate tests.
- Term-by-term differentiation
- The process of differentiating each term of a convergent power series to form a new series for the derivative.
- Term-by-term integration
- The process of integrating each term of a convergent power series to form a new series for an antiderivative.
Common Mistakes to Avoid
- Forgetting to start the derivative sum at n = 1 is wrong because the n = 0 constant term differentiates to 0.
- Assuming endpoint behavior is unchanged is wrong because differentiating or integrating can make an endpoint series converge or diverge differently.
- Changing the radius of convergence after term-by-term calculus is wrong for power series because the derivative and integral series keep the same radius R.
- Dropping the constant of integration is wrong because an indefinite integral represents a whole family of functions, so + C is required.
Practice Questions
- 1 Given f(x) = sum from n = 0 to infinity of 3x^n with |x| < 1, find a power series for f'(x) and simplify it as a rational function.
- 2 Find a power series for integral 1/(1 + x^2) dx centered at 0 by using 1/(1 + x^2) = sum from n = 0 to infinity of (-1)^n x^(2n), and state the radius of convergence.
- 3 A power series has radius of convergence R = 4 centered at x = 2. Explain why its derivative and integral series also have radius 4, and describe what still must be checked.