Problem 47 - A First Course in Complex Analysis with Applications

Question

In this problem you are guided through the proof of the proposition: If the series \(\sum_{k=1}^{\infty} z_{k}\) converges absolutely, then the series converges. Proof We begin with the hypothesis that \(\sum_{k=1}^{\infty}\left|z_{k}\right|\) converges. If \(z_{k}=a_{k}+i b_{k}\), then \(\sum_{k=1}^{\infty}\left|a_{k}\right| \leq \sum_{k=1}^{\infty} \sqrt{a_{k}^{2}+b_{k}^{2}}=\sum_{k=1}^{\infty}\left|z_{k}\right|\) (a) First, explain why the foregoing inequality is true. Second, explain why this inequality shows that the series \(\sum_{k=1}^{\infty}\left|a_{k}\right|\) converges. (b) Explain how your reasoning in part (a) also shows that \(\sum_{k=1}^{\infty}\left|b_{k}\right|\) converges. (c) Explain how parts (a) and (b) show that \(\sum_{k=1}^{\infty} z_{k}\) converges.

Step-by-Step Solution

Answer

The series converges because the absolute convergence implies both real and imaginary parts converge, hence the complex series converges.

Step 1: Understanding the Initial Inequality

The inequality \( \left|a_k\right| \leq \sqrt{a_k^2 + b_k^2} \) holds due to the fact that \( \sqrt{a_k^2 + b_k^2} = \left|z_k\right| \), which is the modulus of the complex number \( z_k = a_k + ib_k \). By definition of absolute value, \( \left|a_k\right| \) is less than or equal to the magnitude of the vector formed by \( a_k \) and \( b_k \). Thus, \( \sum_{k=1}^{\infty} \left|a_k\right| \leq \sum_{k=1}^{\infty} \left|z_k\right| \). This shows \( \sum_{k=1}^{\infty} \left|a_k\right| \) converges because it is dominated by the convergent series \( \sum_{k=1}^{\infty} \left|z_k\right| \), which converges absolutely.

Step 2: Applying to Imaginary Parts

Similarly, for the imaginary parts, we have \( \left|b_k\right| \leq \sqrt{a_k^2 + b_k^2} = \left|z_k\right| \). Thus, \( \sum_{k=1}^{\infty} \left|b_k\right| \leq \sum_{k=1}^{\infty} \left|z_k\right| \). Since the right-hand side series converges, \( \sum_{k=1}^{\infty} \left|b_k\right| \) must also converge by the comparison test.

Step 3: Combining Results for Convergence of Series

Having established that both \( \sum_{k=1}^{\infty} \left|a_k\right| \) and \( \sum_{k=1}^{\infty} \left|b_k\right| \) converge, we can now conclude that \( \sum_{k=1}^{\infty} z_k \) converges. A complex series \( \sum_{k=1}^{\infty} z_k \) converges if and only if the series of its real parts \( \sum_{k=1}^{\infty} a_k \) and imaginary parts \( \sum_{k=1}^{\infty} b_k \) both converge. Since we have shown the absolute convergence for both parts separately, the entire series \( \sum_{k=1}^{\infty} z_k \) converges.

Key Concepts

Absolute Convergence

Absolute convergence is a key concept in the world of complex analysis. It refers to a series whose terms, when replaced by their absolute values, form a convergent series. To understand this better, consider a series \( \sum_{k=1}^{\infty} z_{k} \) where each \( z_{k} \) is a complex number, expressed as \( z_{k} = a_{k} + i b_{k} \). The test for absolute convergence then involves the series \( \sum_{k=1}^{\infty} |z_{k}| \). If this series of absolute values \( \sum_{k=1}^{\infty} \sqrt{a_{k}^2 + b_{k}^2} \) converges, then the original series is said to converge absolutely.

This is an important criterion because if a series converges absolutely, it automatically implies that the original series \( \sum_{k=1}^{\infty} z_{k} \) converges as well. This is more powerful than just having conditional convergence, where the series might converge but not in its absolute term form. Thus, absolute convergence assures a more stringent form of convergence, making it vital in proofs and analysis.

Complex Series

Complex series are series where each term is a complex number, and such series play a crucial role in complex analysis. When dealing with complex series like \( \sum_{k=1}^{\infty} z_{k} \), it's valuable to break down each complex number into its real and imaginary parts: \( z_{k} = a_{k} + i b_{k} \). These series can then be explored through the convergence of their real parts \( \sum_{k=1}^{\infty} a_{k} \) and imaginary parts \( \sum_{k=1}^{\infty} b_{k} \).

The concept of absolute convergence (discussed in earlier sections) facilitates the analysis of complex series, allowing us to apply tests originally designed for real-valued series to complex ones. Essentially, you can treat the series of the real and imaginary parts separately, securing their convergence through absolute values, which leads to the overall convergence of the complex series. This approach is invaluable for solving problems and proving the convergence of sequences and series in the complex plane.

Comparison Test

The comparison test is a handy tool in determining whether a series converges. It's especially useful when dealing with non-trivial series where direct computation of limits might be cumbersome. When applying the comparison test to complex series, you often deal with their absolute values.

Here's how it works: If you have two series, \( \sum_{k=1}^{\infty} a_{k} \) and \( \sum_{k=1}^{\infty} b_{k} \), where \( |a_{k}| \leq |b_{k}| \) for all \( k \), and the series \( \sum_{k=1}^{\infty} b_{k} \) is known to converge, then the series \( \sum_{k=1}^{\infty} a_{k} \) also converges. With complex series, by comparing the absolute terms \( |a_{k}| \) with a known convergent series, we can easily establish convergence.

In the context of our exercise, the Comparison Test helped us confirm that both the series of real parts and the imaginary parts converge. Since the absolute convergence of \( \sum_{k=1}^{\infty} |z_k| \) ensures the parts separately converge, the entire complex series \( \sum_{k=1}^{\infty} z_k \) converges. The comparison test simplifies the checking process of convergence and provides confidence in the results obtained.