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Math101learn.math101.caRadius of Convergence
A rigorous, example-driven guide to radius of convergence, including hypotheses, method choice, verification, and practice.
The central idea
The radius of convergence $R$ for $\sum c_n(x-a)^n$ is the boundary distance between absolute convergence and divergence: $|x-a|<R$ converges and $|x-a|>R$ diverges. Often $R=1/\lim|c_{n+1}/c_n|$ when that limit exists, or $R=1/\limsup|c_n|^{1/n}$.
Definitions, hypotheses, and notation
Termwise differentiation multiplies coefficients by roughly $n$, and integration divides by roughly $n$. Taking nth roots makes these polynomial factors tend to one, which is why the radius remains unchanged. Endpoints can change because polynomial factors matter when the geometric part has magnitude one.
If the coefficient ratio limit does not exist, the limsup root formula supplies the general radius. In routine examples, factorials suggest the ratio test and expressions raised to the $n$th power suggest the root test. Choose the form that isolates $|x-a|$ cleanly.
Conceptual meaning
Coefficient growth competes with the geometric factor $(x-a)^n$. Rapidly growing coefficients shrink the allowed radius; rapidly decaying coefficients enlarge it. The radius says nothing by itself about convergence exactly on the boundary.
A dependable method and decision rule
- Identify coefficient and center.
- Apply the ratio or root test to the absolute term.
- Solve the resulting strict inequality in $|x-a|$.
- Read $R$ as the positive distance from center to either boundary.
- Treat $R=0$, finite $R$, and $R=\infty$ distinctly, then test endpoints only if an interval is requested.
Fully worked example
Graphical or geometric meaning
On the real line the open convergence region is centered at $a$ with symmetric radius $R$. In the complex plane it is a disk; singularities of the represented function often determine the distance to its nearest boundary.
Common mistakes and why they fail
Verification and reasonableness checks
- Verify both provisional endpoints are distance $R$ from $a$.
- Test a simple point such as the center, which must converge.
- Compare coefficient growth with the magnitude of the claimed radius.
Radius is a distance from the center
The radius $R$ gives absolute convergence for $|x-c|<R$ and divergence for $|x-c|>R$; points with $|x-c|=R$ remain undecided. A ratio or root test often yields $|x-c|L<1$, so $R=1/L$ with the usual zero and infinite conventions. Radius is nonnegative and is not the interval's right endpoint. Finite initial terms and constant multiples do not alter it, but scaling the variable does. After finding $R$, test $c-R$ and $c+R$ independently. A response giving only the radius is incomplete when an interval is requested, and an interval without its center visible is prone to a translation error.
Practice
- Find $R$ for $\sum(x/2)^n$.
- Find $R$ for $\sum n!x^n$.
- Find $R$ for $\sum x^n/n!$.
Answers and brief solutions
- $2$.
- $0$.
- $\infty$.
Connections and next steps
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
What is the radius of convergence of Σ n(x−2)^n/5^n?
- Absolute convergence requires |x−2|/5<1.
- Thus |x−2|<5.
- The radius is 5.
End of lesson
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