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Chapter 1: Problem 16

Assume the series \(\sum a_{n}\) is absolutely convergent, and let \(a:=\sum_{n=0}^{\infty} a_{n} .\) Suppose 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\()\)

Short Answer

Expert verified
The series \( \sum c_n \) converges, and \( c = ab \).

Step by step solution

01

Understand the Given Information

We have two series: \( \sum a_{n} \) which is absolutely convergent, implying \( \sum |a_{n}| \) also converges, and \( \sum b_{n} \) which is convergent. We wish to establish that the series \( \sum c_{n} \), defined by \( c_{n} = \sum_{u=0}^{n} a_{u} b_{n-u} \), is convergent and that its sum is \( c = ab \), where \( a = \sum_{n=0}^{\infty} a_{n} \) and \( b = \sum_{n=0}^{\infty} b_{n} \).
02

Use Definitions of Series

Consider the partial sums. Let \( A_N = \sum_{n=0}^{N} a_{n} \) and \( B_N = \sum_{n=0}^{N} b_{n} \). The given conditions tell us that as \( N \to \infty \), \( A_N \to a \) and \( B_N \to b \).
03

Expand the Definition of \( c_n \)

The term \( c_n \) is expanded as \( c_n = a_0 b_n + a_1 b_{n-1} + \cdots + a_n b_0 \). We recognize this as a form of convolution of the sequences \( (a_n) \) and \( (b_n) \).
04

Prove Convergence Using Absolute Convergence

Since \( \sum a_{n} \) is absolutely convergent, we know \( \sum |a_{n}| \) converges. The series \( \sum b_{n} \) is convergent as given. By the absolute convergence of \( \sum a_{n} \), we can rearrange its terms without affecting convergence. Using convolution properties and the fact that rearrangements are allowed, \( c_n \) aligns with the conditions of Mertens' Theorem.
05

Apply Mertens' Theorem

Mertens' Theorem for convolutions states that if one series is absolutely convergent and the other is convergent, then the convolution of these series is convergent. Therefore, \( \sum c_{n} \) is convergent.
06

Compute the Sum of \( \sum c_n \)

By applying the linchpin result of Mertens' Theorem:\( c = \lim_{N \to \infty} (A_N B_N) = ab \). This satisfies \( c = ab \).
07

Conclusion

The steps verify that the series \( \sum c_{n} \) is not only convergent but also the sum equals the product \( a \cdot b \), as per the theorem.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Absolute Convergence
Absolute convergence is an important concept in the study of series. It refers to a series \( \sum a_n \) where the series of absolute values, \( \sum |a_n| \), also converges. This means that the series not only converges, but its terms do not "wander" too far from zero. When a series is absolutely convergent, it can be rearranged without affecting its sum. This property is especially useful in multiple areas of mathematics, such as when applying Mertens' Theorem.

In practice, this implies:
  • If \( \sum |a_n| \) converges, so does \( \sum a_n \).
  • Reordering the terms of an absolutely convergent series does not alter their sum.
Absolute convergence is stronger than simple convergence because it guarantees stability in terms of series summation even under reordering. This concept plays a critical role when dealing with the convolution of two series, ensuring the convergence of their combined series.
Exploring Series Convergence
Series convergence focuses on the behavior of series as the number of terms approaches infinity. A series \( \sum a_n \) is said to converge if its sequence of partial sums has a finite limit. This means the series settles to a particular value, rather than growing unbounded.

Key properties include:
  • If the series converges to a sum \( S \), the difference between the partial sum \( S_N \) and \( S \) approaches zero as \( N \) increases.
  • An important type of convergence is conditional convergence, where the series converges but the series of absolute values \( \sum |a_n| \) diverges.
In the context of the exercise, understanding convergence is pivotal. While the series \( \sum a_n \) describes absolute convergence, \( \sum b_n \) in the exercise is merely convergent. However, thanks to Mertens' Theorem, we can still make robust predictions about their convolution's behavior.
Convolution of Series Simplified
The convolution of series is a mathematical operation that combines two sequences to form a new sequence. For two series \( \sum a_n \) and \( \sum b_n \), their convolution \( c_n \) is defined as the sum \( c_n = \sum_{u=0}^n a_u b_{n-u} \). This creates a new series \( \sum c_n \), capturing the interaction between the two original series.

Mertens' Theorem uses convolution principles to assert that:
  • If \( \sum a_n \) is absolutely convergent and \( \sum b_n \) is convergent, then \( \sum c_n \) is also convergent.
  • The sum of the new series \( \sum c_n \) equals the product of the sums of the original series. This means \( c = ab \), where \( a \) and \( b \) are the sums of \( \sum a_n \) and \( \sum b_n \) respectively.
This concept highlights the elegant interplay between convergence properties and operations like convolution in series analysis. It's a powerful tool that makes it easier to handle complex series through their simpler components.

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Most popular questions from this chapter

Where does the function \(f: \mathbb{C}^{\bullet} \rightarrow \mathbb{C}, f(z)=z \bar{z}+z / \bar{z}\), satisfy the CAUCHYRIEMANN differential equations?

Determine, in each case, all the \(z \in \mathbb{C}\) with $$ \begin{aligned} \exp (z) &=-2, & \exp (z) &=\mathrm{i}, & & \exp (z) &=-\mathrm{i} \\ \sin z &=100, & \sin z &=7 \mathrm{i}, & \sin z &=1-\mathrm{i} \\ \cos z &=3 \mathrm{i}, & \cos z &=3+4 \mathrm{i}, & \cos z &=13. \end{aligned} $$

Let \(\operatorname{Maps}\left(\mathbb{N}_{0}, \mathbb{C}\right)\) be the set of all maps of \(\mathbb{N}_{0}\) into \(\mathbb{C}(=\) the set of all complex number sequences). Show: The map $$ \begin{aligned} \sum: \operatorname{Maps}\left(\mathbb{N}_{0}, \mathbb{C}\right) & \longrightarrow \operatorname{Maps}\left(\mathbb{N}_{0}, \mathbb{C}\right) \\ \left(a_{n}\right)_{n} \geq 0 & \longmapsto\left(S_{n}\right)_{n \geq 0} \text { with } S_{n}:=a_{0}+a_{1}+\cdots+a_{n} \end{aligned} $$ is bijective the (telescope trick). The theories of sequences and of infinite series are therefore in principle the same.

Find the real and imaginary parts of each of the following complex numbers: $$ \frac{\mathrm{i}-1}{\mathrm{i}+1} ; \quad \frac{3+4 \mathrm{i}}{1-2 \mathrm{i}} ; \quad \mathrm{i}^{n}, n \in \mathbb{Z} ; \quad\left(\frac{1+\mathrm{i}}{\sqrt{2}}\right)^{n}, n \in \mathbb{Z} $$ $$ \left(\frac{1+\mathrm{i} \sqrt{3}}{2}\right)^{n}, n \in \mathbb{Z} ; \quad \sum_{\nu=0}^{7}\left(\frac{1-\mathrm{i}}{\sqrt{2}}\right)^{\nu} ; \frac{(1+\mathrm{i})^{4}}{(1-\mathrm{i})^{3}}+\frac{(1-\mathrm{i})^{4}}{(1+\mathrm{i})^{3}} $$

Let \(D=\mathbb{R}\) or \(D=\mathbb{C} .\) Let \(C \in \mathbb{C}\) be a constant and \(f: D \rightarrow \mathbb{C}\) differentiable with $$ f^{\prime}(z)=C f(z) \quad \text { for all } \quad z \in D $$ If \(A=f(0)\), then $$ f(z)=A \exp (C z) \quad \text { for all } \quad z \in D $$
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