When students learn about convergence tests in Calculus II, they sometimes make mistakes. These mistakes can lead to wrong conclusions about whether a series converges (gets closer to a specific value) or diverges (doesn't settle on a specific value). Understanding these common errors is important for mastering this topic. Here are some frequent mistakes and why it's good to avoid them.

• Ignoring the Type of Series: Each test for convergence works best with certain types of series. For example, the geometric series test only works when the series looks like $a r^n$. If you use the wrong test, like the ratio test on a polynomial series, you might get the wrong answer.

• Using the Ratio Test Wrongly: The ratio test helps us understand series. It says that for a series $\sum a_n$, if you find the limit $L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$ and:

  • If $L < 1$, the series converges.
  • If $L > 1$ (or $L = \infty$), the series diverges.
  • If $L = 1$, you can't decide.

  It's very important to know that if $L$ is 1, you should try another test. Saying a series converges or diverges just because the ratio equals 1 can lead to mistakes.

• Forgetting Absolute Convergence: A common mistake is thinking that if $\sum a_n$ converges, then $\sum |a_n|$ must also converge. This isn’t always true. For example, the alternating harmonic series converges, but the harmonic series diverges. Always check if the convergence is absolute (which means $\sum |a_n|$ converges) or conditional.

• Misusing the Comparison Test: When you use the comparison test, you need to compare a series $\sum a_n$ with another series $\sum b_n$. Make sure:

  • If $0 \leq a_n \leq b_n$ for all $n$, and $\sum b_n$ converges, then $\sum a_n$ also converges.
  • If $0 \leq b_n \leq a_n$ for all $n$, and $\sum a_n$ diverges, then $\sum b_n$ also diverges.

  A frequent mistake is not applying these conditions correctly, especially the inequality parts. Always check that your comparisons are correct for all terms.

• Mixing Up p-Series Rules: Remember that a p-series $\sum \frac{1}{n^p}$ only converges if $p > 1$. A common error is thinking it converges just because of an intuitive guess instead of looking at the exponent.

• Not Checking for Divergence: It's just as important to check if a series diverges. Sometimes it might not be easy to see if a series diverges. For example, if you use the limit comparison test incorrectly, you might miss that a series can diverge, even when compared to another diverging series.

• Ignoring the First Few Terms: The convergence of infinite series really starts to matter only after a certain point. The first few terms usually don't affect convergence, but they can lead to mistakes in calculations. Always focus on the limit as $n$ gets very large.

• Using Only One Test: It can be easy to rely on just one test to make conclusions. If the result isn't clear, try multiple tests. Switching between the ratio test and root test can help you find clarity on complex series.

• Being Inconsistent with Limits: When calculating limits for tests, be careful with your math. Simple mistakes, like forgetting L'Hôpital's Rule or not simplifying correctly, can lead to big errors.

By being aware of these common mistakes when using convergence tests, students can improve their understanding and accuracy when working with series. Careful application of definitions, proper comparisons, and using different tests will help build confidence in their conclusions. A careful approach reduces errors and helps students understand series convergence better, which is crucial in calculus.