Power series
Objective. Find radii and intervals of convergence.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
A power series is an infinite series of the form Σ cₙ(x − a)ⁿ, which behaves like a polynomial of infinite degree centred at a. Every power series has a radius of convergence R, found typically via the ratio test applied to the general term, such that the series converges absolutely for |x − a| < R and diverges for |x − a| > R. The interval of convergence is this range together with a separate check of the two endpoints x = a − R and x = a + R, since behaviour there must be tested individually (often with comparison or alternating tests). For example, Σ xⁿ/n has R = 1 by the ratio test, converges at x = −1 by the alternating series test, but diverges at x = 1 since it becomes the harmonic series.
Worked example
Find the radius of convergence of the power series Σ (from n=1 to ∞) of xⁿ/n².
- 1Apply the ratio test to aₙ = xⁿ/n²: compute |aₙ₊₁/aₙ| = |x|·n²/(n+1)².
- 2Take the limit as n → ∞: n²/(n+1)² → 1.
- 3The ratio test limit is |x|, so the series converges when |x| < 1.
- 4Read off the radius of convergence directly from this inequality.
Answer: R = 1
Concept mastery
Based on your past attempts at this lesson — the question types to practise again come first.
Complete a practice set to see your concept breakdown here.
Practice
Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.
Start practisingCommon mistake. Students find the radius R correctly but forget to test both endpoints separately, assuming convergence or divergence holds uniformly on the closed interval; each endpoint needs its own dedicated test.
Your progress
Tick each step as you finish it. Your exact reading position is saved automatically, so Continue drops you back on the same line.
0 of 5 steps complete
Your place saves automatically as you read.