Power Series — Radius of Convergence & Examples

A power series is an infinite sum built from powers of $(x-c)$:

Here $c$ is the center and the constants $a_n$ are the coefficients. In almost every problem the real task is the same: find the values of $x$ for which the series converges.

When You Run the Convergence Procedure

You apply this method whenever you are handed a power series and asked where it converges — which covers most exam and homework prompts, plus the analysis behind Taylor and Maclaurin series. Convergence is organized by the radius of convergence $R$:

• it converges when $|x-c| < R$,
• it diverges when $|x-c| > R$,
• the boundary case $|x-c| = R$ must be tested separately.

The radius is a distance from the center, not a set of $x$-values. For real-variable problems that distance becomes the interior interval

with the endpoints possibly included or not. This matters because inside the interval you can treat the series like a very long polynomial — differentiating, integrating, and approximating term by term are justified there, but not automatically everywhere.

The Steps

1. Identify the center $c$ by writing the series as $\sum a_n (x-c)^n$.
2. Find the radius $R$ using the ratio test (or root test) on $|x-c|$.
3. Write the interior interval $(c-R,\, c+R)$.
4. Test each endpoint separately by substituting it in.

Full Worked Example: Find the Radius and Interval

Consider

This is centered at $c=2$. Apply the ratio test to $a_n = \frac{(x-2)^n}{3^n}$:

Convergence requires

so the radius is

giving the interior interval $(-1,5)$. Now test the endpoints one at a time.

At $x=5$:

which diverges.

At $x=-1$:

which also diverges, because its terms alternate between $1$ and $-1$ instead of approaching $0$.

So the final interval of convergence is

That is the full workflow in one example: identify the center, find $R$, write the inside interval, then test both endpoints.

Where Each Step Goes Wrong, and How to Check

• Mixing up radius and interval. The radius is a number like $R=3$; the interval is a set like $(-1,5)$. Related, but different objects.
• Forgetting the center $c$. With $(x-2)^n$, the distance test uses $|x-2|$, not $|x|$. Re-read the exponent before testing.
• Skipping the endpoint tests. The ratio and root tests usually say nothing at the endpoints, so check them individually after finding $R$.
• Assuming both endpoints behave the same. Even with the same radius, one endpoint may converge while the other diverges; the behavior depends on the series you get after substitution.

Try a Similar Series

Work through

Find the center, solve for the radius, and test the endpoints. Then look at a Taylor series and notice the same convergence ideas reappear — the procedure does not change, only the coefficients do.