When it shows a series converges, which of the following tests will also show the series converges absolutely? Select all that are correct.
Test for Divergence

Geometric Series Test

Integral Test

Direct Comparson Test

Limit Comparison Test

Alternating Series Test

Ratio Test

Root Test

To determine if a series converges absolutely, we need a test that confirms the convergence of the series when all terms are taken as positive, regardless of their original sign. Here’s a breakdown of each test with regard to absolute convergence:

1. **Test for Divergence**: This test is used to determine if a series diverges by checking if the terms do not tend to zero. It does not confirm absolute convergence.
   - **Not useful for absolute convergence.**

2. **Geometric Series Test**: The geometric series test shows absolute convergence for any converging geometric series with a common ratio \( |r| < 1 \).
   - **Useful for absolute convergence**.

3. **Integral Test**: This test can show absolute convergence, as it applies to series whose terms are positive, and it establishes convergence for such terms by comparison with an improper integral.
   - **Useful for absolute convergence**.

4. **Direct Comparison Test**: This test is useful for showing absolute convergence because it compares a series term-by-term with a known convergent series.
   - **Useful for absolute convergence**.

5. **Limit Comparison Test**: This test also compares the terms of a series to those of a known convergent series, establishing absolute convergence if it succeeds.
   - **Useful for absolute convergence**.

6. **Alternating Series Test**: This test only shows conditional convergence, not absolute convergence. Therefore, it is not useful for absolute convergence.
   - **Not useful for absolute convergence.**

7. **Ratio Test**: This test is very effective in determining absolute convergence, especially when terms are positive.
   - **Useful for absolute convergence**.

8. **Root Test**: This test can show absolute convergence if the limit condition holds.
   - **Useful for absolute convergence**.

### Summary

The following tests **can show absolute convergence**:
- **Geometric Series Test**
- **Integral Test**
- **Direct Comparison Test**
- **Limit Comparison Test**
- **Ratio Test**
- **Root Test**

The **Test for Divergence** and the **Alternating Series Test** do not confirm absolute convergence.

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Would you like more details on any of these tests or have any questions? 

### Additional Related Questions

1. What is the difference between absolute and conditional convergence?
2. Can absolute convergence imply convergence, and if so, why?
3. How does the ratio test differ in effectiveness compared to the root test?
4. When is the integral test most effective for determining convergence?
5. Why doesn’t the Alternating Series Test show absolute convergence?

### Tip

When deciding on a convergence test, try to simplify the series terms as much as possible; some tests work best on specific forms.

When it shows a series converges, which of the following tests will also show the series converges absolutely? Select all that are correct.
Test for Divergence
Geometric Series Test
Integral Test
Direct Comparison Test
Limit Comparison Test
Alternating Series Test
Ratio Test
Root Test

Explore which convergence tests can confirm the absolute convergence of a series. Understand the applicability of tests like the Geometric Series Test, Integral Test, and Ratio Test in determining absolute convergence. Suitable for advanced high school and college calculus students.

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