False
Absolute Convergence vs Convergence
- Absolute convergence refers to the convergence of the series obtained by taking the absolute values of the terms of the original series.
- Convergence, on the other hand, simply means that the series converges, without considering the signs of the terms.
Counterexample
- It is important to note that not every absolutely convergent series is convergent.
- A classic counterexample is the Alternating Harmonic Series: 1 - 1/2 + 1/3 - 1/4 + ...
- This series is conditionally convergent (convergent but not absolutely convergent) because the series of absolute values diverges, while the original series converges.
Importance
- Understanding the distinction between absolute convergence and convergence is crucial in the study of series, as it has implications for the properties and behavior of series.
- Absolute convergence is a stronger condition than convergence, as it implies convergence but not vice versa.
Conclusion
- In conclusion, the statement that every absolute convergent series is convergent is false, as demonstrated by the counterexample of the Alternating Harmonic Series.
- It is essential to grasp the nuances of absolute convergence and convergence to fully comprehend the convergence properties of series.