Finding the radius of convergence for power series can be tricky for students. There are some common mistakes that often lead to confusion or wrong answers. Let’s break down these issues and make them easier to understand.

One big mistake is not using the Ratio Test or Root Test correctly. These tests are really important for figuring out the radius of convergence, which we can call $R$, for a series that looks like $\sum a_n x^n$. A frequent error happens when students don’t pay close attention to the limit. For example, when using the Ratio Test, some forget to take the absolute value of the ratio. This can mess up the results about whether the series converges or not.

Another common problem is forgetting to check the endpoints of the interval of convergence. After finding the radius $R$, students sometimes think that what happens at the endpoints $x = -R$ and $x = R$ is the same as what happens inside the interval. This can be misleading. It’s important to test what happens at both endpoints to see if the series converges or diverges there.

Also, remember that the radius of convergence shows how far the series reaches from its center, but it doesn’t tell you the full interval. Some students mistakenly think the interval of convergence is just $(-R, R)$. In reality, it can be open, closed, or half-open depending on what you find out at the endpoints. This needs to be checked carefully.

Students also often forget that not all power series start at $x=0$. While most examples show this case, series that are centered at a different point, like $x = c$, need a different way to check for convergence. The formula for the radius of convergence stays the same; it’s just the center that changes. Students should write their series correctly based on this center to help them understand it better and get the right answers.

Finally, there’s a mistake that comes from relying too much on calculators or computer tools without understanding how they work. These tools can quickly show if a series converges, but it’s really important to know the steps behind these calculations. Students should engage with the ideas behind the math to ensure they can solve similar problems on their own later.

To avoid these common mistakes when finding the radius of convergence, students should:

1. Carefully use the Ratio and Root Tests, including absolute values.
2. Check the endpoints separately to see if they converge or diverge.
3. Understand that the interval of convergence might not just be $(-R, R)$.
4. Look out for a center point that isn’t zero when checking convergence.
5. Work on understanding the concepts instead of just relying on tools.

By paying attention to these points, students can get better at handling power series in calculus.