A power series is like a polynomial with infinitely many terms, and it usually converges only for x-values near its center x = a. The radius of convergence R tells how far left and right from a the series is guaranteed to converge. The interval of convergence gives the complete set of x-values where the series converges, including any endpoints that work.
This matters because functions represented by power series are only equal to those series inside their intervals of convergence.
The Ratio Test is one of the most common tools for finding R because power series terms often contain powers, factorials, or exponential patterns. For a series sum c_n(x - a)^n, the Ratio Test studies the limit L = lim as n goes to infinity of |u_(n+1)/u_n|. The series converges when L < 1 and diverges when L > 1, which usually produces an inequality involving |x - a|.
After finding the open interval, each endpoint must be tested separately because the Ratio Test is often inconclusive when L = 1.
Key Facts
- A power series has the form sum c_n(x - a)^n, where a is the center.
- The radius of convergence R is the distance from the center a to either end of the convergence region.
- Use the Ratio Test with L = lim as n goes to infinity |u_(n+1)/u_n|.
- The series converges when L < 1 and diverges when L > 1.
- If the Ratio Test gives |x - a| < R, the open interval is (a - R, a + R).
- Endpoints x = a - R and x = a + R must be checked by substituting each value into the original series.
Vocabulary
- Power series
- An infinite series of the form sum c_n(x - a)^n, centered at x = a.
- Center
- The value a in a power series, which is the midpoint of the interval of convergence.
- Radius of convergence
- The nonnegative number R that gives the distance from the center to the edge of the convergence interval.
- Interval of convergence
- The full set of x-values for which a power series converges.
- Ratio Test
- A convergence test that uses the limit of the absolute value of consecutive term ratios to decide whether a series converges.
Common Mistakes to Avoid
- Forgetting to check endpoints. The Ratio Test usually finds only the open interval, and the endpoint behavior can be different on the left and right.
- Using the simplified inequality as the final interval too soon. First solve for |x - a| < R, then translate that into an interval centered at a.
- Testing endpoints in the ratio limit instead of the original series. At endpoints the Ratio Test often gives L = 1, so you must substitute the endpoint into the original power series and use another test.
- Assuming both endpoints behave the same way. One endpoint may create an alternating series while the other creates a p-series or harmonic series, so each endpoint needs its own check.
Practice Questions
- 1 Find the radius and interval of convergence for sum from n = 1 to infinity of (x - 2)^n/n.
- 2 Find the radius and interval of convergence for sum from n = 0 to infinity of n!(x + 1)^n/5^n.
- 3 A Ratio Test calculation gives convergence for |x - 3| < 4 and is inconclusive when |x - 3| = 4. Explain what values must be checked next and why the Ratio Test alone is not enough.