Difficulty: Intermediate | Prerequisites: Geometric series, ratio test for sequences
A power series is an infinite polynomial in (x - a). It converges on some interval around a, and the radius of convergence tells you how wide that interval is. Finding the radius usually means applying the ratio test, but you must always check the endpoints separately because the ratio test is inconclusive there.
Power series
An infinite series of the form sum of a_n (x - a)^n for n = 0 to infinity, where a is the centre.
Think of it as an infinite polynomial whose convergence depends on x.
Radius of convergence (R)
The number R such that the series converges absolutely for |x - a| < R and diverges for |x - a| > R. R can be zero (converges only at x = a), a positive number, or infinity (converges everywhere).
Interval of convergence (I)
The set of all x-values for which the power series converges. It is the open interval (a - R, a + R) plus whichever endpoints pass their own convergence test.
In simple terms, R tells you the half-width; endpoint checks tell you whether to use square or round brackets on each side.
Ratio test (for series)
Compute L = lim |a_{n+1} / a_n| as n goes to infinity. If L < 1 the series converges absolutely; if L > 1 it diverges; if L = 1 the test is inconclusive.
Geometric series
The series sum of r^n for n = 0 to infinity, which equals 1/(1 - r) when |r| < 1. This is the workhorse for building power series representations of rational functions.
Power series representation (PSR)
Rewriting a function as a power series, typically by manipulating it into a form that matches the geometric series template 1/(1 - r).
The standard procedure:
Write out |a_{n+1} / a_n| for the given series.
Take the limit as n goes to infinity.
Set that limit less than 1 and solve for x to find R.
Then check each endpoint individually with a separate convergence test (AST, limit comparison, p-series, etc.).
Series: sum of (x - 7)^n / (3n)! for n = 0 to infinity.
Apply the ratio test:
|a_{n+1} / a_n| = |(x - 7)^{n+1} / (3(n+1))!| times |(3n)! / (x - 7)^n|
= |x - 7| / [(3n+3)(3n+2)(3n+1)]
As n goes to infinity, this limit equals 0 for all x.
Since the limit is 0 < 1 for every x in the reals, R = infinity. The series converges for all real x.
Series: sum of (x - 2)^n / (4 + 7n) for n = 0 to infinity.
Ratio test:
|a_{n+1} / a_n| = |(x - 2)| times (4 + 7n) / (11 + 7n).
As n goes to infinity, the fraction tends to 1, so the limit is |x - 2|.
Set |x - 2| < 1, giving 1 < x < 3. So R = 1, centre a = 2.
Endpoint x = 1:
The series becomes sum of (-1)^n / (4 + 7n).
The terms alternate, decrease in absolute value, and tend to 0.
By the alternating series test, it converges.
Endpoint x = 3:
The series becomes sum of 1 / (4 + 7n).
By the limit comparison test with sum of 1/n (since (1/(4+7n)) / (1/n) tends to 1/7), this diverges.
Interval of convergence: I = [1, 3), or equivalently 1 <= x < 3.
Find a PSR for f(x) = 14 / (14 + x).
Rewrite to match the geometric template:
f(x) = 14 / (14 + x) = 14 / [14(1 + x/14)] = 1 / (1 + x/14) = 1 / (1 - (-x/14)).
This matches 1/(1 - r) with r = -x/14.
So the PSR is sum of (-x/14)^k for k = 0 to infinity = sum of (-1)^k (x/14)^k.
The geometric series converges when |r| < 1, i.e. |x/14| < 1, i.e. -14 < x < 14.
Interval of convergence: (-14, 14).
Ratio test for power series:
R = \lim_{n \to \infty} \left| \frac{a_n}{a_{n+1}} \right|or equivalently, set the limit of |a_{n+1}/a_n| < 1 and solve for x.
Geometric series:
\sum_{k=0}^{\infty} r^k = \frac{1}{1 - r}, \quad |r| < 1To build a PSR for a rational function, rewrite it so the denominator has the form (1 - r), then expand.
Students often skip endpoint testing. The ratio test only gives the open interval; convergence at the endpoints requires separate work.
Forgetting that R = infinity is a valid answer. If the ratio test limit is 0 for all x, the series converges everywhere.
Confusing the radius of convergence with the interval of convergence. R is a number; I is an interval that may or may not include its endpoints.
When building a PSR, students sometimes forget to rewrite the function so the denominator matches (1 - something) exactly, leading to sign errors.
Radius and interval of convergence is a staple exam question. Expect to apply the ratio test, simplify, and then test two endpoints.
PSR questions usually ask you to rewrite a function as a geometric series and state the interval of convergence.
Pay close attention to whether the problem asks for R (a number) or I (an interval with brackets).
Endpoint convergence tests vary: the alternating series test, limit comparison test, and p-series test are the most common.
True or false: If the ratio test limit equals 0 for all x, the radius of convergence is 0. (False, R = infinity.)
Fill in the blank: The geometric series 1/(1 - r) converges when ___. (|r| < 1.)
True or false: If a power series converges at x = 5 and the centre is a = 2, the radius of convergence must be at least 3. (True.)
Fill in the blank: After finding R, you must test the ___ separately. (Endpoints.)
True or false: The interval of convergence [1, 3) means the series converges at x = 1 but diverges at x = 3. (True.)
Q: Find the radius of convergence for sum of (x - 7)^n / (3n)!.
A: Apply the ratio test. The limit is |x - 7| times 1/[(3n+3)(3n+2)(3n+1)], which tends to 0 for all x. So R = infinity.
Q: Find the interval of convergence for sum of (x - 2)^n / (4 + 7n).
A: Ratio test gives |x - 2| < 1, so R = 1 and the preliminary interval is (1, 3). At x = 1: alternating series test shows convergence. At x = 3: limit comparison with 1/n shows divergence. I = [1, 3).
Q: Write a power series representation for f(x) = 14/(14 + x) and state the interval of convergence.
A: Rewrite as 1/(1 - (-x/14)). PSR = sum of (-x/14)^k = sum of (-1)^k x^k / 14^k. Converges for |x| < 14, so I = (-14, 14).
Q: A power series centred at a = 3 has R = 5. What is the open interval of convergence (before endpoint checks)?
A: (3 - 5, 3 + 5) = (-2, 8).
Power series convergence underpins everything in the Maclaurin/Taylor series notes: those series are power series, so their convergence behaviour follows the same rules. The geometric series PSR technique connects directly to the series-applications notes, where you differentiate or integrate known series to obtain new ones.
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Power series, radius of convergence, interval of convergence, ratio test, root test, geometric series, power series representation, PSR, endpoint convergence, alternating series test, limit comparison test, M 2414, integral calculus, UT Austin