Problem 11 - Advanced Engineering Mathematics

Question

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 \(\lim _{n \rightarrow \infty} \operatorname{Im}\left(z_{n}\right)\). \(\left\\{\frac{4 n+3 n i}{2 n+i}\right\\}\)

Step-by-Step Solution

Answer

The sequence converges to the complex number \(2 + \frac{3}{2}i\).

Step 1: Separate Real and Imaginary Parts

Begin by dividing both the numerator and denominator of the sequence \(z_n = \frac{4n + 3ni}{2n + i}\) by \(n\) to simplify the expression, by separating real and imaginary parts. This gives:\[ z_n = \frac{4 + 3i}{2 + \frac{i}{n}}. \]

Step 2: Compute the Real Part Limit

To find \(\lim_{n \to \infty} \operatorname{Re}(z_n)\), we first expand the expression:\[ z_n = \frac{(4 + 3i)(2 - \frac{i}{n})}{(2 + \frac{i}{n})(2 - \frac{i}{n})} = \frac{(4 + 3i)(2 - \frac{i}{n})}{4 + \frac{1}{n^2}}. \]Simplify the numerator:\[ (4 + 3i)(2 - \frac{i}{n}) = 8 + \frac{4i}{n} + 6i - \frac{3}{n} = 8 + (\frac{4}{n} + 6)i - \frac{3}{n}. \]Thus, the real part is \(\operatorname{Re}(z_n) = \frac{8 - \frac{3}{n}}{4 + \frac{1}{n^2}}\).Taking the limit as \(n \to \infty\), we get:\[ \lim_{n \to \infty} \operatorname{Re}(z_n) = \frac{8}{4} = 2. \]

Step 3: Compute the Imaginary Part Limit

Now find \(\lim_{n \to \infty} \operatorname{Im}(z_n)\). Use the imaginary simplification found earlier:\[ \operatorname{Im}(z_n) = \frac{\frac{4}{n} + 6}{4 + \frac{1}{n^2}}. \]Taking the limit as \(n \to \infty\), the terms involving \(\frac{1}{n}\) disappear:\[ \lim_{n \to \infty} \operatorname{Im}(z_n) = \frac{6}{4} = \frac{3}{2}. \]

Step 4: Conclude the Limit of the Sequence

Since we now have both limits, the sequence \(z_n\) converges to the complex number \(L = 2 + \frac{3}{2}i\). This conclusion comes from the real and imaginary limits we computed.

Key Concepts

Complex Numbers

A complex number is a fundamental concept in mathematics, particularly in complex analysis. It is expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. The imaginary unit \(i\) is defined by the property \(i^2 = -1\).

Complex numbers provide a way to solve equations that have no real number solutions. They are essential in various fields such as engineering, physics, and applied mathematics. Complex numbers also enable us to explore more advanced mathematical concepts like complex analysis, which involves studying functions that operate in the complex plane.

• The real part of a complex number is \(a\).
• The imaginary part is \(b\).
• Complex numbers can be represented graphically in the complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part.

Real and Imaginary Parts

In the given sequence \(\left\{z_{n}\right\}\), each term is a complex number comprised of real and imaginary components. By expressing a complex number as \(z = x + yi\), the real and imaginary parts can be distinguished:

• \(\operatorname{Re}(z_n)\) denotes the real part of the sequence terms.
• \(\operatorname{Im}(z_n)\) stands for the imaginary part of the terms.

To handle complex sequences, it often helps to separate these parts to simplify calculations, such as determining limits as \(n\) approaches infinity. The complexity of expressions may appear to escalate, but this separation allows for a more manageable evaluation of real and imaginary contributions separately. In this exercise, we deconstructed \(z_n\) to compute real and imaginary limits separately before concluding the overall convergence of the sequence.

Limits of Sequences

Limits are a critical concept in calculus and analysis, representing the value that a sequence or function approaches as the index or input grows indefinitely. Computing limits of sequences involving complex numbers requires careful handling of both real and imaginary components.

For the sequence \(\{z_n\}\), the goal is to determine the limits of both \(\operatorname{Re}(z_n)\) and \(\operatorname{Im}(z_n)\) separately. This requires simplifying the sequence, often starting by dividing both the real and imaginary components by \(n\), which helps in finding behavior as \(n\) becomes large.
The sequence convergence is confirmed if both the real and imaginary parts approach definite numbers.

Sequence Convergence

Sequence convergence, in the context of complex numbers, implies that the values of the sequence approach a specific complex number as the index \(n\) increases. For sequence \(\{z_n\}\), it converges to \(L = 2 + \frac{3}{2}i\) because both its real part, \(\operatorname{Re}(z_n)\), and imaginary part, \(\operatorname{Im}(z_n)\), reach respective limits:

• \(\lim_{n \to \infty} \operatorname{Re}(z_n) = 2\)
• \(\lim_{n \to \infty} \operatorname{Im}(z_n) = \frac{3}{2}\)

The analysis involves separating the sequence into its components and computing each part's limit, indicating convergence to a complex number. In general, for any sequence of complex numbers to converge, both its real and imaginary series must converge independently.