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Interval of Convergence

Level 1 - Math I (Physics) topic page in Power Series.

Principle

The interval of convergence is the full set of real \(x\)-values where a power series converges. The radius gives the open interval around the centre, but the endpoints need separate tests.

Open interval from radius
\[c-R\lt x\lt c+R\]

Endpoint behaviour matters because many useful series converge at one endpoint but not the other. The geometric series and logarithm series show this clearly.

Notation

\(c\)
centre of the series
\(R\)
radius of convergence
\(c-R\)
left endpoint
\(c+R\)
right endpoint
\(I\)
interval of convergence

The interval may be open, closed, or half-open depending on endpoint tests.

Method

Step 1: Find the radius

Use a convergence test, usually the ratio test, to get \(|x-c|\lt R\). This gives the open interval.

Step 2: Test the left endpoint

Substitute \(x=c-R\) into the original series. Do not test the inequality again; it will give equality and no decision.

Left endpoint
\[x=c-R\]
Shifted value
\[x-c=-R\]
Endpoint series
\[\sum_{n=0}^{\infty}a_n(-R)^n\]
Decision
\[\text{apply an ordinary series test}\]

Step 3: Test the right endpoint

Substitute \(x=c+R\):

Right endpoint
\[x=c+R\]
Shifted value
\[x-c=R\]
Endpoint series
\[\sum_{n=0}^{\infty}a_nR^n\]
Decision
\[\text{apply an ordinary series test}\]

Rules

Possible open interval
\[(c-R,c+R)\]
Possible closed interval
\[[c-R,c+R]\]
Possible half-open intervals
\[[c-R,c+R)\quad\text{or}\quad(c-R,c+R]\]

If \(R=\infty\), the interval of convergence is all real numbers. If \(R=0\), only the centre may converge.

Examples

Question
Find the interval of convergence of
\[\sum_{n=0}^{\infty}x^n\]
Answer
The radius condition is
\[|x|\lt1\]
so the open interval is
\[(-1,1)\]
At
\[x=1\]
the series is
\[1+1+1+\cdots\]
which diverges. At
\[x=-1\]
the terms alternate
\[1,-1,1,-1,\ldots\]
and do not approach zero in partial sums. The interval is
\[(-1,1)\]

Checks

  • Always test endpoints by substitution into the original series.
  • The ratio test usually gives no answer at endpoints.
  • The interval of convergence includes exactly the convergent endpoints.
  • Radius and interval are related, but they are not the same object.

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Radius of Convergence

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Taylor Polynomials