Problem 16 - Complex Analysis

Question

Assume that the series \(\sum a_{n}\) is absolutely convergent, and let \(a:=\sum_{n=0}^{\infty} a_{n}\) Suppose that the series \(\sum b_{n}\) is convergent, and assume \(b:=\sum_{n=0}^{\infty} b_{n-}\) Show: if \(c_{n}:=\sum_{\nu=0}^{n} a_{\nu} b_{n-\nu}\), then the series \(\sum c_{n}\) is convergent, and for \(c:=\sum_{n=0}^{\infty} c_{n}\) one has \(c=a b \quad\) (MERTENS' theorem, F. MERTENS, 1875\()\)

Step-by-Step Solution

Answer

The series \( \sum c_n \) is convergent and its sum is \( c = ab \).

Step 1: Understand the Given Series

We are given two convergent series: \( \sum a_n \), which is absolutely convergent, and \( \sum b_n \). We denote their sums by \( a \) and \( b \) respectively. Our task is to show that the series \( \sum c_n \) with terms \( c_n = \sum_{u=0}^{n} a_{u} b_{n-u} \) is convergent, and that its sum is \( ab \).

Step 2: Use the Cauchy Product Formula

The terms \( c_n = \sum_{u=0}^{n} a_{u} b_{n-u} \) are characteristic of the Cauchy product of two series. According to the Cauchy product theorem, if one series is absolutely convergent and the other is convergent, then the Cauchy product series converges as well. Therefore, we conclude that \( \sum c_n \) is convergent.

Step 3: Apply Mertens' Theorem

Mertens' theorem further tells us that if \( \sum a_n \) is absolutely convergent and \( \sum b_n \) is convergent, then the Cauchy product series \( \sum c_n = \sum (\sum_{u=0}^{n} a_{u} b_{n-u}) \) not only converges but also its sum \( c = \sum_{n=0}^{\infty} c_n \) equals the product of the two series' sums, \( ab \).

Step 4: Arrive at Conclusion

By applying Mertens' theorem, since \( \sum a_n \) is absolutely convergent and \( \sum b_n \) is convergent, we have \( c = ab \). This means that the sum of the series \( \sum c_n \) is indeed \( ab \), confirming the assertion in the exercise.

Key Concepts

Absolute Convergence

When we talk about absolute convergence, we refer to a very specific type of series convergence. Consider a series \( \sum a_n \) to be absolutely convergent if the series of absolute values \( \sum |a_n| \) is convergent. This is a stronger condition than just being convergent.
One might wonder why absolute convergence is so important. Here's why:

• If a series is absolutely convergent, it is also convergent. This is not necessarily true for the converse, making absolute convergence a more robust form of convergence.
• It ensures that the series behaves well under rearrangements. That is, rearranging the terms of an absolutely convergent series doesn't affect its convergence or its sum.

This property plays a crucial role in the context of the Cauchy product and Mertens' theorem, ensuring that products of series behave predictably.

Cauchy Product

The Cauchy product is a way to multiply two infinite series together. Given two series \( \sum a_n \) and \( \sum b_n \), their Cauchy product is the series \( \sum c_n \), where the terms are given by:
\[ c_n = \sum_{u=0}^{n} a_{u} b_{n-u} \]

• In general, the Cauchy product is used in contexts where we need to understand the behavior of a product of functions.
• It's essential when the individual series are not absolutely convergent, as its convergence behavior needs to be carefully examined.

Mertens' theorem provides exactly the conditions under which this Cauchy product converges. It tells us that if one series is absolutely convergent and the other is simply convergent, their product will converge and its sum will be the product of the sums of the original series.

Convergent Series

A series is convergent if the sequence of its partial sums approaches a finite limit. For the series \( \sum a_n \) to be convergent, the limit
\[ \lim_{n \to \infty} S_n = a \]
must exist, where \( S_n \) is the sum of the first \( n \) terms.
Convergent series are foundational in mathematics because:

• They provide a way to deal with infinite processes by summarizing them with a finite number.
• This idea is central to many areas like analysis, number theory, and applied sciences.

In our problem context, knowing that both \( \sum a_n \) and \( \sum b_n \) are convergent helps apply Mertens' theorem, specifically leveraging the absolute convergence of \( \sum a_n \) to ensure the series \( \sum c_n \) also converges.

Complex Analysis

Complex analysis is a field dealing with complex numbers and the functions that operate on them. Complex functions can often be represented as power series, highlighting the importance of convergence.
One of the power tools in complex analysis is the study of series like our problem series. These represent functions as sums of terms coefficiented by powers of a variable. Key areas where this is important:

• Working within the world's famous integrals like contour integrals.
• Solving otherwise complex differential equations.

In complex analysis, understanding the convergence of series is crucial because it directly impacts the validity of using series representations of functions. Through series, we can explore many properties of complex functions, making the convergence properties discussed in the exercise incredibly important in this field.