Finding the Interval of Convergence for Power Series

When we talk about power series, we want to know where they work best. This is really important in calculus because it tells us which numbers we can safely use without causing problems.

What is a Power Series?
A power series is an infinite series that looks like this:

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

Here, $a_n$ are numbers called coefficients, and $c$ is the center of the series. Our goal is to figure out which $x$ values make this series work well.

Radius of Convergence
To find the interval of convergence, we usually start by figuring out the radius of convergence, which we call $R$.

We can do this using the Ratio Test or the Root Test.

The Ratio Test tells us to look at a limit, which is like checking the series as we go further out:

$$L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$$

If $L$ is a number we can have, then:

• The series works well if $L < 1$.
• The series does not work well if $L > 1$.

This gives us the expression:

$$|x - c| < R$$

where $R = \frac{1}{L}$.

We can also use the Root Test which looks like this:

$$L = \limsup_{n \to \infty} \sqrt[n]{|a_n|}$$

The conclusion is similar: the series works well when $|x - c| < \frac{1}{L}$.

Finding the Interval of Convergence
Once we find $R$, we can write the interval of convergence as:

$$(c - R, c + R)$$

But we must check if the series also works at the ends of this range, $c - R$ and $c + R$.

Checking the Endpoints
For both endpoints, we plug in the values and check the series:

1. For the left endpoint, where $x = c - R$:

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

   We can use tests like the p-series test, integral test, or the Alternating Series Test to see if this series works.

2. For the right endpoint, where $x = c + R$:

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

   We check this using similar tests.

Final Interval
After we finish checking the endpoints, here’s what the interval of convergence can look like:

• $(c - R, c + R)$ - (neither endpoint included)
• $[c - R, c + R]$ - (both endpoints included)
• $(c - R, c + R]$ - (left endpoint not included, right endpoint included)
• $[c - R, c + R)$ - (left endpoint included, right endpoint not included)

In summary, understanding the interval of convergence is super important for power series. It helps us know where to use these series safely and confidently in calculus!