Calculating sums of series, especially geometric and telescoping series, can be tough for students in a Calculus II class. It's important to understand these ideas, but there are some common mistakes to watch out for. Here are some of the errors you should avoid:

1. Confusing Series with Sequences
A big mistake students make is mixing up series and sequences. A sequence is just a list of numbers, while a series is what you get when you add those numbers together. For example, the sequence $a_n = \frac{1}{n}$ is different from the series $\sum_{n=1}^{\infty} a_n$. When you are working with a series, you are adding the terms, not just looking at them one by one.

2. Not Identifying the Type of Series
Before you try to sum a series, it’s super important to know what type it is. For geometric series, the formula is $S = a \cdot \frac{1 - r^n}{1 - r}$ when $|r| < 1$. Here, $a$ is the first term, and $r$ is called the common ratio. If you use a formula for one type of series on another type, you’re going to get the wrong answer. Always figure out what kind of series you’re working with.

3. Misusing the Geometric Series Formula
You can only use the geometric series formula in specific cases. Make sure the common ratio $r$ is less than 1 in absolute value ($|r| < 1$) for it to work. If you use this formula when $|r| \geq 1$, your answer will be wrong, and you might miss that the series doesn't actually add up!

4. Forgetting About Partial Fraction Decomposition
For telescoping series, using partial fraction decomposition is often necessary. These series usually get simpler when some terms cancel each other out. For example, take a look at the series

$\sum_{n=1}^{\infty} \left( \frac{1}{n} - \frac{1}{n + 1} \right)$

You’ll see quick cancellation here. If you don’t apply this method, you might end up doing more work and getting it wrong. Always simplify and see how the terms fit together.

5. Ignoring Convergence Tests
Before you start adding a series, it’s important to check if it converges (adds up) or diverges (doesn't add up). Many students jump right into the numbers without checking this first. You can use tests like the Ratio Test, the Root Test, and the Comparison Test. For example, to check a series like $\sum_{n=1}^{\infty} \frac{1}{n^2}$, the p-test can show it converges. But the series $\sum_{n=1}^{\infty} \frac{1}{n}$ diverges. If you skip this step, you might make wrong guesses about the series.

6. Overlooking Infinite Behavior
When you are looking at the sum of an infinite series, it’s key to understand what happens to the terms as they get really big. For example, in a geometric series, if $r \geq 1$, the terms will keep getting bigger or jump around, which shows it diverges. Don’t skip checking the limit of the series as $n$ goes to infinity; this can give you important clues.

7. Assuming Absolute Convergence
Not all series that converge do so in a straightforward way. It’s crucial to know the difference between conditional and absolute convergence, especially with alternating series. A series that converges conditionally may not converge if you add up the absolute values instead. For instance, the alternating harmonic series converges, but the harmonic series does not. This difference is important for making sure your calculations are correct.

8. Forgetting to Use Limits Properly
When you’re working with series that involve limits, make sure to handle those limits carefully. Messing up with limits can lead you to misunderstand the sums. For example, when calculating

$\lim_{n \to \infty} S_n$

where $S_n = a_1 + a_2 + \ldots + a_n$, make sure you calculate the limit correctly to really reflect how the series behaves.

By paying attention to these common mistakes and avoiding them, students can get better at calculating sums of series, especially geometric and telescoping series. Going through Calculus II can be challenging, but being aware of these pitfalls can make the experience more successful and rewarding.