The interval of convergence is an important idea when working with power series. It tells us the range of values where the series adds up to a specific number.

A power series usually looks like this:

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

Here, $a_n$ are the numbers in front of each term, $c$ is the center point of the series, and $(x - c)$ is the part that changes. To find the interval of convergence, we first need to figure out the radius of convergence, which is represented by $R$. This radius is the distance from the center $c$ to the nearest point where the series does not work.

We often use the Ratio Test or Root Test to find this radius.

Knowing the interval of convergence is not just helpful for understanding where the series works but also because power series can describe functions within their intervals. If a power series adds up at a certain point in its interval, you can differentiate (find the slope) and integrate (find the area under the curve) it term by term. This is really useful in calculus!

The formula for finding the radius of convergence is:

$$\frac{1}{R} = \limsup_{n \to \infty} |a_n|^{1/n} \text{ or } \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right|$$

Once we know $R$, we can say the interval of convergence is from $(c-R)$ to $(c+R)$. But, it's very important to check the endpoints, $c-R$ and $c+R$, to see if the series converges there because it might not even though it works in the middle.

Let’s look at an example with the power series for $f(x) = \sum_{n=0}^{\infty} x^n$. This series converges when $|x| < 1$. So, the radius of convergence is $R = 1$, meaning the interval is $(-1, 1)$. However, if we check the endpoints:

• At $x = -1$, we get the series $\sum_{n=0}^{\infty} (-1)^n$, which doesn’t add up to a number (it diverges).
• At $x = 1$, we have $\sum_{n=0}^{\infty} 1$, which also doesn’t add up.

So, the interval of convergence stays $(-1, 1)$. This shows how important it is to check the endpoints carefully.

In short, the interval of convergence tells us where the power series works and what function it might represent. This is important for understanding calculus and math as a whole. Knowing this interval helps us use math operations and theorems correctly within that range.