To find out if a series is absolutely convergent or conditionally convergent, we look at a series that looks like this: (\sum a_n).

Step 1: Check for Absolute Convergence

• What It Means: A series (\sum a_n) is called absolutely convergent if the series made of its absolute values (\sum |a_n|) converges.

• How to Check: We can use tests like the Ratio Test, Root Test, or Comparison Test on (\sum |a_n|). If this series converges, then (\sum a_n) is absolutely convergent too.

Step 2: Check for Conditional Convergence

• What It Means: If the series (\sum a_n) converges, but the series of its absolute values (\sum |a_n|) does not, then (\sum a_n) is called conditionally convergent.

• How to Check: After we see that (\sum |a_n|) does not converge, we use tests like the Alternating Series Test on (\sum a_n) to check if it converges.

Step 3: Quick Summary of the Tests

• If (\sum |a_n|) converges, then (\sum a_n) is absolutely convergent.

• If (\sum a_n) converges and (\sum |a_n|) does not converge, then (\sum a_n) is conditionally convergent.

• If both (\sum a_n) and (\sum |a_n|) do not converge, then the series is divergent.

Examples and Special Cases

• Example of Absolute Convergence: Take the series (\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n^2}). We can find that (\sum_{n=1}^{\infty} \frac{1}{n^2}) converges. So, (\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n^2}) converges absolutely.

• Example of Conditional Convergence: The series (\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n}) converges using the Alternating Series Test, but (\sum_{n=1}^{\infty} \frac{1}{n}) does not converge. This shows that it is conditionally convergent.

It's important to understand the difference between absolute and conditional convergence, especially in advanced math. This knowledge helps us work with series better, especially when dealing with functions and integration.