Power Series: Step-by-Step Guide
A power series centered at c has the form sum c_n (x - c)^n. It converges for x in an interval centered at c, with a radius of convergence R found by the ratio or root test. Inside this interval the series defines a smooth function; on the endpoints convergence must be checked separately.
Math Racer skill guide · Calculus 2 · AP Calc BC / College
How to solve it, step by step
- Identify the form and the center. Write the series as sum c_n (x - c)^n. Identify the coefficient c_n and the center c.\[\sum_{n=0}^{\infty} c_n (x-c)^n\]
- Apply the ratio test for the radius of convergence. Compute L = lim |c_(n+1)/c_n| * |x - c|. Set L < 1 and solve for |x - c| < R.\[\lim_{n\to\infty} \left| \frac{c_{n+1}}{c_n} \right| |x-c| < 1\]
- Determine the radius of convergence R. R is the value such that the series converges for |x - c| < R and diverges for |x - c| > R.\[R = \lim_{n\to\infty} \left| \frac{c_n}{c_{n+1}} \right|\]
- Test the endpoints. Substitute x = c - R and x = c + R into the original series. Use a series convergence test (alternating, p-series, comparison) to decide each endpoint.\[\text{Test } x = c \pm R \text{ separately}\]
- State the interval of convergence. Combine the open interval (c - R, c + R) with whichever endpoints converge to write the full interval.\[I = [c-R,\, c+R],\; (c-R, c+R],\; \dots\]
Worked example
Find the radius and interval of convergence of sum from n=1 to infinity of (x - 2)^n / n.
Step 1: The series is centered at c = 2 with c_n = 1/n.
Step 2: Apply the ratio test: lim |c_(n+1)/c_n| * |x - 2| = lim (n/(n+1)) * |x - 2| = |x - 2|. Set |x - 2| < 1.
Step 3: Radius of convergence R = 1, interior interval 1 < x < 3.
Step 4: Test endpoints. At x = 3: sum 1/n is the harmonic series, which diverges. At x = 1: sum (-1)^n / n is the alternating harmonic series, which converges.
Step 5: Interval of convergence is [1, 3).
Verification: This is a shifted version of the standard log series, whose convergence on [-1, 1) is well-known.
Tips and common mistakes
- The ratio test almost always gives the radius; don't try other tests first for power series.
- When applying the ratio test, factor |x - c| out of the limit.
- If the limit of |c_(n+1)/c_n| is 0, the series converges for all x (R = infinity).
- If the limit is infinity, the series only converges at x = c (R = 0).
- Endpoints must be tested individually with a separate convergence test.
- A power series may converge at both endpoints, neither, or just one - all four cases occur.
- Inside the interval of convergence, you may differentiate or integrate term by term.
- Be careful with the form (x - c)^n vs x^n; the center c shifts everything.