Problem 44 - Calculus: One and Several Variables

Question

Show that if \(\sum a_{2}\) is absolutely convergent and \(\left|b_{k}\right| \leq\left|a_{k}\right|\) for at \(J k,\) then \(\sum b_{k}\) is absolutely convergent.

Step-by-Step Solution

Answer

Since \(\sum \left| a_k \right|\) is absolutely convergent and \(\left|b_{k}\right| \leq\left|a_{k}\right|\) for all \(k\), by the Comparison Test, it follows that the series \(\sum b_k\) and \(\sum |b_k|\) must also be convergent. Thus, \(\sum b_{k}\) is absolutely convergent.

Step 1: Find the absolute values of the series

Since we are dealing with an absolutely convergent series, we first need to find the absolute values of both series, that is: \[\sum\left|a_k\right|\] and \[\sum\left|b_k\right|\]

Step 2: Apply the Comparison Test

Given the inequality \(\left| b_{k} \right| \leq \left| a_{k} \right|\) for all \(k\), we can now apply the Comparison Test: Since \(\sum \left| a_k \right|\) is absolutely convergent and \(\left|b_{k}\right| \leq\left|a_{k}\right|\) for all \(k\), by the Comparison Test, it follows that the series \(\sum b_k\) must also be convergent, which means that \(\sum b_k\) is also absolutely convergent since it implies that also \(\sum |b_k|\) converges.

Step 3: Conclusion

Based on Steps 1 and 2, we have now shown that if \(\sum a_{k}\) is absolutely convergent and \(\left| b_{k} \right| \leq \left| a_{k} \right|\) for at \(J k\), then the series \(\sum b_{k}\) is absolutely convergent.

Key Concepts

Comparison Test

The Comparison Test is a powerful tool used to determine the convergence of series. It involves comparing the series in question with another series that is already known to be convergent or divergent. Here's how it works:

If you have two series, say \( \sum a_k \) and \( \sum b_k \), and you know that \( |b_k| \leq |a_k| \) for all terms \(k\) beyond a certain point, then the behavior of \( \sum b_k \) can be inferred from \( \sum a_k \). This is especially useful when \( \sum a_k \) is a series we understand well.

Applying to absolute convergence, if \( \sum |a_k| \) is a known absolutely convergent series, and we have \( |b_k| \leq |a_k| \) for all \(k\), then by the Comparison Test, it follows that \( \sum |b_k| \) is also convergent. This implies that \( \sum b_k \) is absolutely convergent. Thus, the Comparison Test allows transferring the convergence property from one series to another under given inequalities.

Convergent Series

A convergent series is a series whose terms approach a specific number as more terms are added, meaning that the series sum approaches a finite limit. For example, the series \( \sum a_k \) is said to be convergent if the sequence of partial sums \( S_N = a_1 + a_2 + \cdots + a_N \) gets arbitrarily close to some number \( L \) as \( N o \infty \).

In the case of absolute convergence, a series \( \sum a_k \) is absolutely convergent if the series of absolute values \( \sum |a_k| \) is convergent. This is a stronger condition than regular convergence.

• When we have absolute convergence, any rearrangement of the series will also converge to the same limit.
• Absolute convergence guarantees the series is consistent and stable in terms of its limiting behavior.

Understanding these concepts helps evaluate more complex series, using known convergent series as benchmarks.

Inequality in Series

Inequality plays a critical role in comparing series to judge their convergence properties through tests like the Comparison Test. For two sequences \( \{a_k\} \) and \( \{b_k\} \), knowing the inequality \( |b_k| \leq |a_k| \) for each term \(k\) provides foundational information to apply the Comparison Test.

Here's why the inequality is essential:

• The inequality helps establish an upper bound for the sequence \( \{b_k\} \) by comparing it to the sequence \( \{a_k\} \), which possesses known convergence properties (e.g., absolute convergence in \( \sum |a_k| \)).
• It directs us to assess whether the sum of series \( \sum b_k \) might inherit convergence properties from the better-known series \( \sum a_k \).

A firm grasp of analyzing inequalities enables us to judiciously apply mathematical tests and derive conclusions about the series being studied. This approach empowers us to unlock insights into seemingly complex problems by leveraging simpler, understood scenarios.