Interval and Radius of Convergence

students, imagine you are trying to build a function from an infinite recipe 🍰. Each new term gives a better approximation, but only if the recipe behaves well. In Calculus 2, power series help us write functions as infinite sums, and the big question is: for which values of $x$ does the series actually work? That question leads to the radius of convergence and the interval of convergence.

What a Power Series Is and Why Convergence Matters

A power series is an infinite sum of the form

$$\sum_{n=0}^{\infty} a_n(x-c)^n$$

where $a_n$ is a sequence of coefficients, $c$ is the center of the series, and $x$ is the input. A power series looks a lot like a polynomial, except it has infinitely many terms.

For example, the geometric series

$$\sum_{n=0}^{\infty} x^n = 1+x+x^2+x^3+\cdots$$

is a power series centered at $c=0$. It converges when $|x|<1$ and diverges when $|x|\ge 1$. That simple example shows the main idea: a power series does not usually work for every $x$. Instead, it works on a specific range of values.

This range is called the interval of convergence. The size of that interval is controlled by the radius of convergence. These ideas matter because many Calculus 2 topics depend on them, including finding series representations of functions and differentiating or integrating series term by term.

Radius of Convergence: The Basic Picture

For any power series

$$\sum_{n=0}^{\infty} a_n(x-c)^n,$$

there is a number $R$ called the radius of convergence such that one of three things happens:

• The series converges for all $x$ with $|x-c|<R$.
• The series diverges for all $x$ with $|x-c|>R$.
• At the boundary points $x=c-R$ and $x=c+R$, the series may converge or diverge, and each endpoint must be checked separately.

This means the power series behaves like a circle of validity around the center $c$. In one real-variable setting, that “circle” becomes an interval on the number line.

If $R=3$ and the center is $c=2$, then the series converges for all $x$ such that

$$|x-2|<3,$$

which is the interval

$$-1<x<5.$$

That open interval is the starting point. The endpoints $x=-1$ and $x=5$ are not automatically included. They must be tested one at a time.

A special case is $R=0$, where the series converges only at the center. Another special case is $R=\infty$, where the series converges for every real number $x$.

How to Find the Radius of Convergence

The most common tools for finding $R$ are the Ratio Test and the Root Test. These tests work especially well for power series because the variable $x$ appears in each term.

Suppose we have

$$\sum_{n=0}^{\infty} a_n(x-c)^n.$$

Using the Ratio Test, we examine

$$\lim_{n\to\infty}\left|\frac{a_{n+1}(x-c)^{n+1}}{a_n(x-c)^n}\right|=\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right||x-c|.$$

If this limit is less than $1$, the series converges. If it is greater than $1$, the series diverges.

That usually leads to an inequality like

$$|x-c|<R,$$

where $R$ is the radius of convergence.

Example

Consider

$$\sum_{n=1}^{\infty}\frac{(x-4)^n}{n3^n}.$$

Apply the Ratio Test to the terms $u_n=\frac{(x-4)^n}{n3^n}$:

$$\left|\frac{u_{n+1}}{u_n}\right|=\left|\frac{(x-4)^{n+1}}{(n+1)3^{n+1}}\cdot\frac{n3^n}{(x-4)^n}\right|=\left|\frac{x-4}{3}\right|\cdot\frac{n}{n+1}.$$

Now take the limit as $n\to\infty$:

$$\lim_{n\to\infty}\left|\frac{u_{n+1}}{u_n}\right|=\left|\frac{x-4}{3}\right|.$$

The series converges when

$$\left|\frac{x-4}{3}\right|<1,$$

so

$$|x-4|<3.$$

This means the radius of convergence is

$$R=3.$$

At this stage we know the open interval of convergence is

$$1<x<7.$$

Interval of Convergence: Testing the Endpoints

The radius of convergence tells you the interior of the interval, but the interval of convergence includes only the values of $x$ for which the series actually converges. That means endpoint testing is essential.

For the example above, we check $x=1$ and $x=7$.

Endpoint $x=1$

Substitute $x=1$ into the series:

$$\sum_{n=1}^{\infty}\frac{(1-4)^n}{n3^n}=\sum_{n=1}^{\infty}\frac{(-3)^n}{n3^n}=\sum_{n=1}^{\infty}\frac{(-1)^n}{n}.$$

This is the alternating harmonic series, which converges.

Endpoint $x=7$

Substitute $x=7$:

$$\sum_{n=1}^{\infty}\frac{(7-4)^n}{n3^n}=\sum_{n=1}^{\infty}\frac{3^n}{n3^n}=\sum_{n=1}^{\infty}\frac{1}{n}.$$

This is the harmonic series, which diverges.

So the interval of convergence is

$$[1,7).$$

This example shows an important fact: the interval of convergence can be open, closed, half-open, or even a single point. The radius alone does not tell the full story.

Why Endpoints Must Be Checked Separately

students, the reason endpoint behavior is special is that convergence tests often depend on whether a series is strictly inside the radius $R$ or exactly on the boundary. Inside the radius, a power series behaves predictably. On the boundary, the series may turn into a different type of series, such as a geometric series, $p$-series, alternating series, or something more complicated.

For instance, a series may converge at one endpoint because the terms alternate in sign, but diverge at the other endpoint because the terms stay positive and sum too slowly. This is why every endpoint needs its own test.

A good strategy is:

1. Find $R$ using the Ratio Test or Root Test.
2. Write the open interval $|x-c|<R$.
3. Test each endpoint directly with a familiar convergence test.
4. State the final interval of convergence in interval notation.

Connection to the Bigger Picture of Power Series

Interval and radius of convergence are not isolated ideas. They are the foundation for using power series to represent functions.

For example, the geometric series

$$\frac{1}{1-x}=\sum_{n=0}^{\infty}x^n$$

works only when $|x|<1$. If you want to represent another function using a series, you often start by transforming a known power series. But that transformation changes the interval of convergence, so you must track where the new series is valid.

This becomes especially important when differentiating and integrating power series. A major result in Calculus 2 is that, inside the interval of convergence, a power series can be differentiated or integrated term by term. However, the radius of convergence stays the same after differentiation or integration, while the behavior at endpoints can change.

That means if a series converges on $(-2,2)$, its derivative and integral series also have radius $2$, but you still need to check whether the endpoints remain included.

Another Example with a Different Center

Consider

$$\sum_{n=0}^{\infty}\frac{(x+1)^n}{2^n}.$$

This is a geometric series with ratio

$$\frac{x+1}{2}.$$

A geometric series converges when the absolute value of the ratio is less than $1$:

$$\left|\frac{x+1}{2}\right|<1.$$

So

$$|x+1|<2,$$

which gives

$$-3<x<1.$$

Thus the radius of convergence is

$$R=2,$$

and the interval of convergence is initially the open interval $(-3,1)$.

Now check endpoints:

• At $x=-3$, the series becomes

$$\sum_{n=0}^{\infty}\frac{(-2)^n}{2^n}=\sum_{n=0}^{\infty}(-1)^n,$$

which diverges.

• At $x=1$, the series becomes

$$\sum_{n=0}^{\infty}1,$$

which also diverges.

So the interval of convergence is

$$(-3,1).$$

This example shows that sometimes neither endpoint works.

Conclusion

Interval and radius of convergence answer the key question for power series: where does the infinite sum actually represent a valid function? The radius of convergence $R$ tells how far from the center $c$ the series converges automatically inside the region $|x-c|<R$. The interval of convergence gives the complete set of $x$ values, including any endpoints that also converge.

In Calculus 2, these ideas are essential because they control when power series can be used safely, when they can be differentiated or integrated term by term, and how functions can be represented by infinite sums. Mastering convergence means understanding both the interior of the interval and the special behavior at the endpoints. ✅

Study Notes

• A power series has the form $$\sum_{n=0}^{\infty}a_n(x-c)^n.$$
• The radius of convergence $R$ tells how far from the center $c$ the series converges inside the region $$|x-c|<R.$$
• The interval of convergence is the set of all $x$ values where the series converges, including any endpoints that pass separate tests.
• Use the Ratio Test or Root Test to find the radius of convergence.
• After finding the open interval from $|x-c|<R,$ always test the endpoints individually.
• The interval of convergence may be open, closed, half-open, or a single point.
• A power series can often be differentiated and integrated term by term within its interval of convergence.
• The radius of convergence usually stays the same after differentiation or integration, but endpoint behavior may change.
• Geometric series are a key reference example: $\sum_{n=0}^{\infty}r^n$ converges when $$|r|<1.$$
• Understanding convergence is essential for representing functions by series in Calculus 2.