Math101 · Independent Ontario learning library
Math101Radius 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.
