Introduction
A power-series calculation can be numerically correct at each step and still give the wrong conclusion if the center, reciprocal, or endpoint conditions are mishandled. These errors are especially easy to make because the radius and the interval are related, but they are not the same thing.
This tutorial focuses on how to check that reasoning. The central questions are: What distance is being measured? What quantity does the root test actually produce? And what can the radius tell us about convergence, without making claims it does not justify?
Start with the Center
A power series centered at \(c\) has the form
The expression raised to the \(n\)th power is \(x-c\), so convergence is governed by the distance \(|x-c|\) from \(x\) to the center \(c\). Only when \(c=0\) is that distance simply \(|x|\).
The interval \((c-R,c+R)\), when \(0<R<\infty\), is therefore a candidate convergence interval: all its interior points converge, while points outside its closure diverge. Whether either endpoint belongs to the actual convergence set must be considered separately.
Worked Example: Measuring from a Nonzero Center
Consider the series \(\sum_{n=0}^{\infty}2^{-n}(x-10)^n\). Its coefficients are \(a_n=2^{-n}\), so
Thus \(R=1/L=2\), and the distance condition is \(|x-10|<2\), not \(|x|<2\). For example, at \(x=9\), \(|9-10|=1<2\), and the series becomes
This geometric series converges because its ratio has absolute value \(1/2<1\). The incorrect condition \(|x|<2\) would reject \(x=9\), since \(|9|=9\); that rejection is wrong because it measures from zero rather than from the center. The check is direct: at \(x=9\), the \(n\)th term is \((-1/2)^n\), whose absolute value tends to zero and whose geometric series converges.
What the Root Test Measures
For \(n\geq 1\), apply the root test to the terms \(a_n(x-c)^n\). The relevant quantities are
If \(x\ne c\), the factor \(|x-c|\) is fixed and positive. Taking the limit superior gives \(L|x-c|\), where \(L=\limsup_{n\to\infty}|a_n|^{1/n}\). This identity is used only when \(x\ne c\): it avoids the undefined expression \(0\cdot\infty\) that could arise if \(L=\infty\) and \(x=c\).
The root test says that a series converges absolutely if the limit superior of the \(n\)th roots of the absolute terms is less than \(1\), and diverges if that limit superior is greater than \(1\). At equality, the root test gives no conclusion. The radius is obtained from the threshold \(L|x-c|=1\), not by mistaking \(L\) for the radius.
Worked Example: Do Not Reverse the Root-Test Limit
Consider \(\sum_{n=0}^{\infty}4^n x^n\). Here \(a_n=4^n\), so
The radius is the reciprocal of \(L\), not \(L\) itself. At \(x=1/8\), the root-test quantity for the terms is
so the series converges absolutely. Indeed, its terms are \(4^n(1/8)^n=(1/2)^n\), a convergent geometric series. At \(x=1/2\), the quantity is \(4(1/2)=2>1\), so the series diverges; its terms are \(2^n\), which do not tend to zero. These checks agree with \(R=1/4\). Using \(4\) as the radius would give the opposite and incorrect scale.
Why the Radius Formula Works
The formula follows from applying the root test at each point. The center must be handled separately, since the fixed factor \(|x-c|\) is zero there. For every other point, the limit superior is exactly \(L|x-c|\).
For \(x\ne c\), the root-test quantity is \(\limsup |a_n|^{1/n}|x-c|=L|x-c|\). Multiplication by the fixed positive number \(|x-c|\) preserves the limit superior, including when \(L=\infty\).
If \(L|x-c|<1\), the series converges absolutely. If \(L|x-c|>1\), it diverges. Equality is inconclusive by the root test.
For \(0<L<\infty\), the equality \(L|x-c|=1\) is equivalent to \(|x-c|=1/L\). Hence \(R=1/L\).
If \(L=0\), every \(x\ne c\) gives root-test quantity \(0\), so the series converges absolutely everywhere and \(R=\infty\). If \(L=\infty\), every \(x\ne c\) gives quantity \(\infty\), so the series diverges there and \(R=0\). At \(x=c\), all terms with \(n\ge1\) vanish, so the series converges.
This reasoning also explains why a radius is a distance, not an interval. For finite positive \(R\), the distance inequality \(|x-c|<R\) can be written \(c-R<x<c+R\). The center and radius determine this open interior region; they do not automatically add its endpoints.
Radius Is Not the Whole Convergence Set
A common mistake is to report \([c-R,c+R]\) as the convergence interval merely because the radius is \(R\). That notation includes both boundary points, but the root test at a boundary of a finite positive radius has limit superior \(LR=1\). The root test is inconclusive there; it has not established convergence or divergence.
Worked Example: Write a Candidate Interval Without Assuming Its Endpoints
For \(\sum_{n=0}^{\infty}(x-1)^n\), the coefficients are \(a_n=1\), and \(L=1\). Thus \(R=1\), and the candidate interior interval is
At an interior point such as \(x=3/2\), the series is \(\sum (1/2)^n\), which converges. At \(x=3\), the terms are \(2^n\), so the series diverges. The radius calculation itself makes no endpoint claim: at \(x=0\) or \(x=2\), the root-test quantity is \(L|x-1|=1\). It is therefore correct, at this stage, to state the candidate interval \((0,2)\) and to leave endpoint membership for a separate test. Closing the interval solely from the value of \(R\) would not be justified.
Zero Radius and the Center
The statement “at a boundary, the root-test quantity is \(1\)” needs an important qualification: it applies to the two boundary points of a finite positive radius. When \(R=0\), the center itself has distance zero, and the root-test calculation away from the center cannot be applied there.
Worked Example: A Zero Radius Still Includes the Center
Consider \(\sum_{n=0}^{\infty} n!(x-c)^n\). For even \(n\), the last \(n/2\) factors in \(n!\) are at least \(n/2\), so
The right-hand side tends to infinity, hence \(L=\limsup (n!)^{1/n}=\infty\), and \(R=0\). At any \(x\ne c\), the root-test quantity is \(L|x-c|=\infty\), so the series diverges. At \(x=c\), however, the \(n\)th term is \(n!0^n=0\) for every \(n\ge1\), and the \(n=0\) term is \(0! =1\). The series therefore sums to \(1\) at its center. In particular, the root-test quantity there is \(0\) for every \(n\ge1\), not \(L|x-c|\): that product would be the undefined expression \(\infty\cdot0\). The center must be checked directly.
A Reliable Error-Checking Routine
Before reporting a radius or interval, check the following points:
- Center: Is the series written with \((x-c)^n\)? Use \(|x-c|\), not \(|x|\), unless \(c=0\).
- Root-test quantity: Have you found \(L=\limsup |a_n|^{1/n}\) and then used the product \(L|x-c|\)?
- Reciprocal: Have you set \(R=1/L\), including the conventions for \(L=0\) and \(L=\infty\)?
- Interior and exterior: For \(0<R<\infty\), convergence is guaranteed when \(|x-c|<R\) and divergence when \(|x-c|>R\).
- Boundary: Have you avoided claiming endpoint behavior from the radius calculation alone?
- Center in the zero-radius case: Have you checked \(x=c\) directly rather than substituting into an expression involving \(0\cdot\infty\)?
These checks separate errors in algebra from errors in interpretation. A correct root-test limit can still produce a wrong answer if its reciprocal is reversed or if the distance is measured from the wrong center.
Check Your Understanding
For each question, identify which quantity or condition must be checked before making a conclusion.
- A power series is centered at \(c=5\) and has radius \(3\). Write the distance condition for its candidate interior interval.
- If \(\limsup |a_n|^{1/n}=6\), what is the radius, and why is \(6\) not the radius?
- For a finite positive radius, what does the root test say at a point whose distance from the center equals the radius?
- Why must the center be checked separately if the radius is zero and \(L=\infty\)?
- What additional work is needed before deciding whether a finite-radius candidate interval includes either endpoint?
