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Math101learn.math101.caInterval of Convergence
A rigorous, example-driven guide to interval of convergence, including hypotheses, method choice, verification, and practice.
The central idea
A power series $\sum c_n(x-a)^n$ converges for $|x-a|<R$ and diverges for $|x-a|>R$, for some radius $R\in[0,\infty]$. The interval of convergence adds the real endpoints $a-R$ and $a+R$ only when separate substitution tests show convergence there.
Definitions, hypotheses, and notation
At every interior point a power series converges absolutely; conditional convergence can occur only at boundary points. The interval may be open, closed, or half-open, but it always has the radius-determined center when $0<R<\infty$. $R=0$ leaves only the center, while $R=\infty$ gives all real numbers and no finite endpoints to test.
Differentiating or integrating a power series preserves its radius but may change endpoint behavior because coefficients acquire factors or divisors involving $n$. Endpoint tests must therefore be redone for the transformed series.
Conceptual meaning
Inside the radius, exponential decay in $(x-a)^n$ dominates coefficient growth; outside, terms fail to be controlled. At the boundary that decisive ratio becomes one, so harmonic, alternating, or other endpoint-specific behavior takes over.
A dependable method and decision rule
- Use the ratio or root test to solve the strict inequality for $x$.
- Identify the center and provisional open interval.
- Substitute the left endpoint into the original series.
- Substitute the right endpoint independently.
- Combine the results using correct brackets and parentheses, with endpoint classifications if useful.
Fully worked example
Graphical or geometric meaning
On the number line, the center $2$ is equidistant from provisional endpoints. Endpoint dots are filled only after separate series tests; symmetry of the radius does not imply symmetric endpoint inclusion.
Common mistakes and why they fail
Verification and reasonableness checks
- Measure both provisional endpoints exactly $R$ from the center.
- Name the endpoint series after substitution.
- Verify interval notation matches inclusion conclusions.
Radius first, endpoints separately
A power series centered at $c$ converges absolutely for $|x-c|<R$ and diverges for $|x-c|>R$. Ratio or root testing usually finds $R$ but gives no automatic answer at $c\pm R$. Substitute each endpoint into the original series and apply an appropriate series test. One endpoint can converge conditionally while the other diverges, so interval notation must treat them independently. Keep the center visible when solving the absolute-value inequality; otherwise a radius can be mistaken for an endpoint. Termwise differentiation and integration retain the open radius, though endpoint behavior may change. Report both the numerical radius and the final interval because they answer different questions about the series.
As a verification, choose one point strictly inside the proposed interval and one strictly outside it. The term behavior and selected convergence test should agree with the claimed radius before endpoint brackets or parentheses are added.
Practice
- Find the interval for $\sum x^n$.
- Find it for $\sum x^n/n^2$.
- Can the ratio test usually decide endpoints?
Answers and brief solutions
- $(-1,1)$.
- $[-1,1]$.
- No.
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 interval of convergence of Σ from n=1 to ∞ of x^n/n?
- For |x|<1 the series converges absolutely.
- At x=1 it is harmonic and diverges.
- At x=−1 it is alternating harmonic and converges, giving [−1,1).
End of lesson
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