Unit 10: Infinite Sequences & Series
Power Series & Interval of Convergence
Finding the radius and interval of convergence of a power series.
Advanced25 min lesson2 min readUpdated August 12, 2026Author not yet attributed
Prerequisites
- Series Convergence Tests
For Which x Does an Infinite Sum of Powers Actually Converge?
Definition — Power Series, Radius, and Interval of Convergence
A power series Σcₙ(x−a)ⁿ is a series with a variable term x. The ratio test, applied with x treated as fixed, produces a condition |x−a| < R that determines the radius of convergence R; the interval of convergence is (a−R, a+R), with each endpoint checked separately since the ratio test is inconclusive exactly there.
Worked Example — Finding a Radius and Interval of Convergence
Find the interval of convergence of Σₙ₌₁^∞ (x−2)ⁿ/(n·3ⁿ). Apply the ratio test: |aₙ₊₁/aₙ| = |x−2|·[n/(n+1)]·(1/3) → |x−2|/3 as n → ∞. Convergence requires |x−2|/3 < 1, so |x−2| < 3: radius R = 3, giving −1 < x < 5 before checking endpoints. At x = −1: the series becomes Σ(−3)ⁿ/(n·3ⁿ) = Σ(−1)ⁿ/n — the alternating harmonic series, which converges. At x = 5: the series becomes Σ3ⁿ/(n·3ⁿ) = Σ1/n — the harmonic series, which diverges. The interval of convergence is [−1, 5).
Tip
The ratio test alone never settles an endpoint — it's inconclusive exactly where the limit equals 1, so both endpoints of a power series' interval always need a separate, direct convergence test.
Common Mistakes
Assuming both endpoints of an interval of convergence behave the same way.
The example above shows one endpoint (x = −1) included and the other (x = 5) excluded — each endpoint substitutes to a genuinely different series and must be checked independently.
Key Takeaways
- The ratio test finds a power series' radius of convergence, R, from where |x−a|/R < 1.
- Each endpoint of the resulting interval must be checked separately with a direct convergence test, since the ratio test is inconclusive there.
Summary
The final lesson uses power series to build polynomial approximations of a function directly from its derivatives.
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