To find the radius of convergence for a power series, let's first understand what a power series is.

A power series looks like this:

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

Here, $a_n$ are numbers that we call coefficients, $c$ is a fixed number called the center of the series, and $x$ is the variable we can change. The radius of convergence, denoted as $R$, shows the range of $x$ values for which the series adds up to a finite number.

The Ratio Test

One popular way to find the radius of convergence is by using the Ratio Test. This test looks at the limit of the ratio of successive coefficients. It tells us:

If we have the series

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

we calculate

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

• If $L < 1$, the series converges (it adds up to a finite number).
• If $L > 1$, it diverges (it doesn’t add up correctly).
• If $L = 1$, we can't make a conclusion.

Now, we can express the radius of convergence $R$ like this:

$$R = \frac{1}{L} \quad \text{(if } L \text{ exists).}$$

To find $R$, you follow these two steps:

1. Calculate $L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$.
2. Then find $R = \frac{1}{L}$.

If $L$ equals zero, that means the radius of convergence $R$ is infinite, meaning the series converges for all $x$. On the other hand, if $L$ is infinite, then $R$ is zero, and the series only converges at the point $c$.

Example of the Ratio Test

Let’s take the series:

$$\sum_{n=0}^{\infty} \frac{x^n}{n!}.$$

Here, the coefficients are $a_n = \frac{1}{n!}$. We can find $L$ by doing the following:

$$L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \left| \frac{\frac{1}{(n+1)!}}{\frac{1}{n!}} \right| = \lim_{n \to \infty} \frac{n!}{(n+1)!} = \lim_{n \to \infty} \frac{1}{n+1} = 0.$$

Since $L = 0$, we find that

$$R = \frac{1}{0} = \infty.$$

This means the series converges for all $x$.

The Root Test

Another way to find the radius of convergence is using the Root Test. This method is helpful for series with powers. The Root Test says:

If

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

then:

• The series converges if $L < 1$.
• The series diverges if $L > 1$.
• The test is inconclusive if $L = 1$.

To find the radius of convergence with the Root Test, we still have

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

Example of the Root Test

Look at the power series:

$$\sum_{n=0}^{\infty} \frac{x^n}{2^n}.$$

Here, $a_n = \frac{1}{2^n}$. Using the Root Test, we calculate

$$L = \limsup_{n \to \infty} \sqrt[n]{\left| \frac{1}{2^n} \right|} = \limsup_{n \to \infty} \frac{1}{2} = \frac{1}{2}.$$

So we have

$$R = \frac{1}{\frac{1}{2}} = 2.$$

This means the series converges when $|x| < 2$.

Interval of Convergence

Finding the radius of convergence is just one part; we also need to know the interval of convergence. This interval shows the range of $x$ values where the series converges.

To find this interval:

1. Identify the endpoints as $c - R$ and $c + R$.
2. Test the convergence at these endpoints by plugging them back into the series.

Checking the Endpoints

For our earlier example where $R = 2$, we have the interval of convergence as:

$$(-2, 2).$$

Now, we check the endpoints:

1. For $x = -2$:

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

   (it doesn't add up correctly, this is known as the harmonic series).

2. For $x = 2$:

   $$\sum_{n=0}^{\infty} \frac{2^n}{2^n} = \sum_{n=0}^{\infty} 1 = \text{diverges.}$$

So, the series only converges in the interval

$$(-2, 2).$$

Conclusion

To sum up, to find the radius of convergence for a power series, we often use the Ratio Test or the Root Test. After finding $R$, the interval of convergence is $(c - R, c + R)$. It is also important to check the endpoints to see if the series converges at those points. This understanding helps to show when a power series converges and connects to other important concepts in math, like Taylor series.