Question

True or False? Justify your answer with a proof or a counterexample. $$ \text { If } \sum_{n=1}^{\infty}\left|a_{n}\right| \text { converges, then } \sum_{n=1}^{\infty} a_{n} \text { converges. } $$

Short answer

True, by the Absolute Convergence Theorem.

Step-by-step solution

Understanding Absolute Convergence

A series \( \sum_{n=1}^{\infty} a_{n} \) is said to be absolutely convergent if the series of its absolute values \( \sum_{n=1}^{\infty} |a_{n}| \) converges. In simpler terms, the absolute convergence of a series implies that the series formed by taking the absolute values of each of its terms also converges.

Applying Absolute Convergence Theorem

According to the Absolute Convergence Theorem, if the series \( \sum_{n=1}^{\infty} |a_{n}| \) converges, then the series \( \sum_{n=1}^{\infty} a_{n} \) also converges. This implies that absolute convergence implies convergence for the original series.

Conclusion

Given that \( \sum_{n=1}^{\infty} |a_{n}| \) converges, we apply the theorem to conclude that \( \sum_{n=1}^{\infty} a_{n} \) must also converge. Hence, the statement is true.

Key concepts

Convergence Theorem

In mathematics, specifically in the study of series, the Absolute Convergence Theorem plays a crucial role in understanding how series behave. This theorem states that if a series of absolute values \( \sum_{n=1}^{\infty} |a_n| \) converges, then the original series \( \sum_{n=1}^{\infty} a_n \) also converges. The logic behind this is that if the series with non-negative terms (absolute values) can settle into a sum, the original series, which might have terms cancelling each other out, will likely converge as well.

Why is this important? When dealing with alternating series or series with both positive and negative terms, it can be challenging to determine convergence. The Absolute Convergence Theorem offers a helpful pathway for determining convergence by simplifying the terms using their absolute values.

Moreover, absolute convergence is a stronger condition than regular convergence. For example, a series might converge without the absolute values converging, but if the absolute values do converge, it's guaranteed that the original series does too. This adds a layer of assurance in our mathematical analysis.

Series Convergence

Series convergence refers to the ability of the sum of an infinite series to approach a finite value as more and more terms are added. This finite value is what we call the limit of the series. There are different types of convergence, but generally, a series \( \sum_{n=1}^{\infty} a_n \) converges if the sequence of its partial sums \( S_k = a_1 + a_2 + ... + a_k \) approaches a specific number as \( k \) becomes very large.

There are several tests to determine series convergence:

• n-th Term Test: If \( \lim_{n \to \infty} a_n eq 0 \), then the series \( \sum a_n \) diverges.
• Geometric Series Test: A geometric series converges if the absolute value of its common ratio is less than 1.
• Alternating Series Test: An alternating series converges if its terms decrease in absolute value and their limit is 0.

Understanding convergence is essential for solving problems in calculus and analysis, allowing mathematicians and scientists to correctly apply infinite series to real-world scenarios.

Counterexample

A counterexample is a specific case that demonstrates a general statement is false. In mathematics, counterexamples are incredibly useful for disproving statements. For the statement considered in the exercise, fortunately for us, the Absolute Convergence Theorem supports its truth, so we do not need a counterexample here.

However, understanding the role of counterexamples is still essential. They help strengthen our understanding by showing the boundaries where certain mathematical rules apply. Knowing that a theorem holds true for all cases without a counterexample reinforces its validity.

While our exercise showed no need for a counterexample due to the truth of the statement, in mathematical exploration, always watch out for scenarios that might challenge initial assumptions. Always check the conditions under which theorems are valid and apply them judiciously.