Problem 12 - Advanced Engineering Mathematics

Question

In Problems 11 and 12, show that the given sequence \(\left\\{z_{n}\right\\}\) converges to a complex number \(L\) by computing \(\lim _{n \rightarrow \infty} \operatorname{Re}\left(z_{n}\right)\) and \(\lim _{n \rightarrow \infty} \operatorname{Im}\left(z_{n}\right)\). $$ \left\\{\left(\frac{1+i}{4}\right)^{n}\right\\} $$

Step-by-Step Solution

Answer

The sequence converges to 0.

Step 1: Identify Sequence and Limit Formulæ

The given sequence is \( z_n = \left(\frac{1+i}{4}\right)^n \). To find the limit as \( n \to \infty \), compute the limits of both the real and imaginary parts.

Step 2: Express in Polar Form

Convert \( \frac{1+i}{4} \) into polar form. The modulus \( r = \sqrt{\left(\frac{1}{4}\right)^2 + \left(\frac{1}{4}\right)^2} = \frac{1}{\sqrt{8}} \) and the argument \( \theta = \tan^{-1}(1) = \frac{\pi}{4} \). Thus, \( \frac{1+i}{4} = \frac{1}{\sqrt{8}} e^{i \frac{\pi}{4}} \).

Step 3: Compute the Power of Polar Form

Raise the polar form to the power \( n \): \( \left(\frac{1}{\sqrt{8}} e^{i \frac{\pi}{4}}\right)^n = \left(\frac{1}{\sqrt{8}}\right)^n e^{i n \frac{\pi}{4}} \).

Step 4: Separate Real and Imaginary Components

Real part: \( \operatorname{Re}(z_n) = \left(\frac{1}{\sqrt{8}}\right)^n \cos\left(n \frac{\pi}{4}\right) \).Imaginary part: \( \operatorname{Im}(z_n) = \left(\frac{1}{\sqrt{8}}\right)^n \sin\left(n \frac{\pi}{4}\right) \).

Step 5: Determine Limits of Real and Imaginary Parts

As \( n \to \infty \), the real and imaginary components involve \( \left(\frac{1}{\sqrt{8}}\right)^n \). Since \( 0 < \frac{1}{\sqrt{8}} < 1 \), this term tends to 0. Thus, both \( \lim_{n \to \infty} \operatorname{Re}(z_n) = 0 \) and \( \lim_{n \to \infty} \operatorname{Im}(z_n) = 0 \).

Step 6: Conclusion about the Limit of Sequence

With both the real and imaginary parts tending to 0, the sequence converges to the complex number \( L = 0 + 0i = 0 \).

Key Concepts

Sequences and Series

In mathematics, sequences and series are fundamental concepts that pave the way to understanding more complex phenomena, such as limits and convergence. A sequence is an ordered list of numbers, each termed as a "term" of the sequence. In our original exercise, the sequence given is \( \left\{ z_n = \left(\frac{1+i}{4}\right)^n \right\} \). Here, every term represents a complex number.Series, on the other hand, involve the summation of a sequence of numbers. For complex sequences, we often analyze them component-wise, which means evaluating the real and imaginary parts separately. To ascertain if a sequence converges to a particular value, evaluating the limit as \( n \to \infty \) for each component helps. Understanding the difference between these components is essential:

• A sequence is about listing numbers in a given order, while a series is about sum of those numbers.
• Convergence occurs when the sequence keeps getting closer to a specific value.

By following these guidelines, one can better understand how sequences behave and how limits are applied to derive conclusions in complex analysis.

Polar Form

Polar form is a way of representing complex numbers that includes both a magnitude and an angle. It's extremely useful for multiplication and powers of complex numbers.To convert a complex number \( z = a + bi \) into polar form, we use:

• The modulus \( r \), which is the distance of the number from the origin in the complex plane, calculated as \( r = \sqrt{a^2 + b^2} \).
• The argument \( \theta \), the angle the line makes with the positive x-axis, found with \( \theta = \tan^{-1}(\frac{b}{a}) \).

For example, in the sequence \( z_n = \left(\frac{1+i}{4}\right)^n \), converting to polar form involves calculating \( r = \frac{1}{\sqrt{8}} \) and \( \theta = \frac{\pi}{4} \). This represents the number \( \frac{1+i}{4} \) in polar terms as \( \frac{1}{\sqrt{8}} e^{i \frac{\pi}{4}} \).Polar form simplifies powers of complex numbers. Raising to a power involves multiplying the modulus and multiplying the angle of the argument by the power, thus streamlining the process for computation.

Limit of a Sequence

The limit of a sequence is a fundamental concept in calculus and complex analysis. It helps to determine what value the sequence approaches as the number of terms n goes to infinity. For a sequence \( \{a_n\} \), we say the limit as \( n \to \infty \) is \( L \) if the terms of the sequence get arbitrarily close to \( L \) as \( n \) increases without bounds.Taking the sequence \( z_n = \left(\frac{1+i}{4}\right)^n \), it involves managing both the real and imaginary components:

• The real part goes as \( \left(\frac{1}{\sqrt{8}}\right)^n \cos(n \frac{\pi}{4}) \)
• The imaginary part involves \( \left(\frac{1}{\sqrt{8}}\right)^n \sin(n \frac{\pi}{4}) \)

Both these expressions contain \( \left(\frac{1}{\sqrt{8}}\right)^n \), which tends to zero as \( n \) increases because it is a fraction less than one. Thus, both the real and imaginary components reduce to zero. This gives us the result that the sequence \( \{z_n\} \) converges to the complex number \( L = 0 \).The understanding of limits aids in evaluating long-term behavior of sequences and determining their convergence in complex analysis.