Question

Show that if a sequence of complex numbers tends to zero, then the sequence of absolute values tends to zero too, and vice versa. Hint: \(a_{n}+i b_{n} \rightarrow 0\) means \(a_{n} \rightarrow 0\) and \(b_{n} \rightarrow 0\).

Short answer

A sequence \( z_n \rightarrow 0 \) implies \( |z_n| \rightarrow 0 \), and \( |z_n| \rightarrow 0 \) implies \( z_n \rightarrow 0 \).

Step-by-step solution

Definition of Convergence for Complex Numbers

If a sequence of complex numbers \( z_n = a_n + i b_n \) tends to 0, it means that the real part \( a_n \) and the imaginary part \( b_n \) of each complex number both tend to 0 as \( n \) tends to infinity. Formally, this is denoted as \( z_n \rightarrow 0 \) when \( a_n \rightarrow 0 \) and \( b_n \rightarrow 0 \).

Definition of Absolute Value for Complex Numbers

The absolute value of a complex number \( z_n = a_n + i b_n \) is given by \( |z_n| = \sqrt{a_n^2 + b_n^2} \). We will need this to understand the behavior of the absolute values of the sequence.

Absolute Value Tends to Zero if Sequence Tends to Zero

Since \( a_n \rightarrow 0 \) and \( b_n \rightarrow 0 \) as \( n \) tends to infinity, it follows that \( a_n^2 \rightarrow 0 \) and \( b_n^2 \rightarrow 0 \). Therefore, \( a_n^2 + b_n^2 \rightarrow 0\). Because the square root function is continuous, \( \sqrt{a_n^2 + b_n^2} \rightarrow \sqrt{0} = 0 \). Hence, \( |z_n| \rightarrow 0 \).

Converse - Sequence Tends to Zero if Absolute Value Tends to Zero

If \( |z_n| \rightarrow 0 \), then by definition \( \sqrt{a_n^2 + b_n^2} \rightarrow 0 \). Since the square root function is continuous and \( \sqrt{x} = 0 \) if and only if \( x = 0 \), it follows that \( a_n^2 + b_n^2 \rightarrow 0 \). This implies that both \( a_n \rightarrow 0 \) and \( b_n \rightarrow 0 \) because squares of real numbers are zero only if the numbers themselves are zero. Hence, \( z_n \rightarrow 0 \).

Key concepts

Complex Numbers

A complex number is a number that can be expressed in the form \(a + ib\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit, satisfying \(i^2 = -1\). Complex numbers are often visualized on a plane called the complex plane, with the horizontal axis representing the real part and the vertical axis representing the imaginary part.
Simply put:

• The real part is \(a\).
• The imaginary part is \(b\).
• Example: \(3 + 4i\) where 3 is the real part and 4 is the imaginary part.

These numbers are highly useful in various fields like engineering, physics, and applied mathematics. Understanding them is essential for studying more advanced topics like sequence convergence in complex analysis.

Sequence Convergence

Sequence convergence in the context of complex numbers means that as we go further and further along in the sequence, the terms get closer and closer to a specific value. For instance, if we have a sequence \((z_n)\), which is a series of complex numbers, we say \(z_n\) converges to 0 if the distance of each \(z_n\) from 0 becomes arbitrarily small as \(n\) increases. Formally,

• For any \(\epsilon > 0\), there exists an integer \(N\) such that for all \(n > N\), \(|z_n| < \epsilon\).

In simpler terms, if the real part \(a_n\) and the imaginary part \(b_n\) both get very close to zero, we say the complex sequence \(z_n = a_n + ib_n\) converges to 0.
This also implies that the distance or \(|z_n|\), the absolute value of the complex number, becomes very small.

Absolute Value in Complex Numbers

The absolute value (or modulus) of a complex number \(z_n = a_n + ib_n\) is given by \(|z_n| = \sqrt{a_n^2 + b_n^2}\). It measures the distance from the origin (0,0) to the point \((a_n, b_n)\) in the complex plane. The absolute value helps us in understanding the magnitude of a complex number regardless of its direction.
Important aspects:

• The absolute value is always non-negative.
• It simplifies to the Euclidean norm in the real numbers when \(b_n = 0\).
• For any complex number, you can think of \(|z_n|\) as the length of the vector from the origin to the point \((a_n, b_n)\) in Cartesian coordinates.

In terms of sequence convergence, if the sequence of absolute values \(|z_n|\) converges to 0, it means that the distances \(|z_n|\) are becoming very small.

Complex Analysis

Complex analysis is the branch of mathematics that deals with complex numbers and functions of complex variables. It explores properties that are unique to complex functions and provides powerful tools for solving real-world problems.
Why it matters:

• It helps in understanding functions that take and return complex numbers.
• It offers insights into convergence of sequences and series of complex numbers.
• It's vital for applications in engineering, physics, and other sciences.

When studying complex analysis, you will encounter concepts like analytic functions, which are functions that are differentiable in the complex plane, and various theorems that generalize real analytic properties to the complex domain. The convergence of complex sequences is one of the foundational ideas needed to master more advanced topics in this field.