Problem 10 - Advanced Engineering Mathematics

Question

Determine whether the given sequence converges or diverges. \(\left\\{e^{1 / n}+2\left(\tan ^{-1} n\right) i\right\\}\)

Step-by-Step Solution

Answer

The sequence converges to \( 1 + \pi i \).

Step 1: Understand the Components of the Sequence

The given sequence is \( \{e^{1/n} + 2(\tan^{-1}n)i\} \). It is a complex sequence with a real part \( e^{1/n} \) and an imaginary part \( 2(\tan^{-1}n)i \). We need to determine the behavior of both parts as \( n \to \infty \).

Step 2: Examine the Real Part

Consider the real part: \( e^{1/n} \). As \( n \to \infty \), \( \frac{1}{n} \to 0 \). Since the exponential function is continuous, \( e^{1/n} \to e^0 = 1 \).

Step 3: Examine the Imaginary Part

Consider the imaginary part: \( 2(\tan^{-1}n)i \). As \( n \to \infty \), \( \tan^{-1}n \to \frac{\pi}{2} \) because \( \tan^{-1} \) approaches \( \frac{\pi}{2} \) for large \( n \). Therefore, the imaginary part \( 2(\tan^{-1}n) \to 2 \times \frac{\pi}{2} = \pi \).

Step 4: Determine the Limit of the Entire Sequence

Combine the results from Steps 2 and 3: as \( n \to \infty \), the sequence \( e^{1/n} + 2(\tan^{-1}n)i \to 1 + \pi i \). Thus, the sequence converges to \( 1 + \pi i \).

Step 5: Conclusion on Convergence

Since both the real and imaginary parts of the sequence converge, the entire sequence converges to a complex number \( 1 + \pi i \).

Key Concepts

Real and Imaginary Parts Analysis

In complex sequences, understanding both the real and imaginary parts is essential. These two components work together to form the behavior of the sequence as a whole.
To explore convergence, analyze each part separately:

• Real Part: This is the section without the imaginary unit, often represented by numbers like a constant or a function affecting its behavior.
• Imaginary Part: This includes terms multiplied by the imaginary unit "i". Often functions like trigonometric functions appear here.

For the sequence \( e^{1/n} + 2(\tan^{-1}n)i \), the real part \( e^{1/n} \) approaches 1 as \( n \to \infty \), diving into the tiny slice near zero where the exponential's subtle nature shows. The imaginary part \( 2(\tan^{-1}n)i \) veers towards \( \pi \), illustrating how inverse trigonometric functions handle large inputs, gliding lovingly towards its ultimate angle limit.
Both analyses are crucial to say definitively whether a sequence converges.

Exponential Function Behavior

The exponential function is a cornerstone of mathematical sequences due to its continuous and smooth nature. It showcases unique behaviors that are extremely useful.
When we look at the component \( e^{1/n} \) from the sequence, we're scrutinizing how an exponential responds when its exponent shrinks towards zero.

• As \( n \to \infty \), \( 1/n \to 0 \).
• This makes \( e^{1/n} \to e^0 = 1 \), gently moving towards a stable and fixed point.

This convergence to a value of 1 is an example of how exponential functions can tame complexity by boiling down to a simple constant. This part of the sequence shows steady behavior, setting the stage for determining whether or not the whole sequence anchors at a point.

Inverse Trigonometric Functions

Inverse trigonometric functions connect angles with ratios, tracing pathways through which large inputs lead to understanding limits. In our sequence, the \( \tan^{-1}n \) terms attract special interest.
As \( n \to \infty \), the behavior of \( \tan^{-1}n \) demonstrates a march towards the angle \( \frac{\pi}{2} \), dictating the stride of our imaginary component.

• \( \tan^{-1} \) squeezes saturation out of infinity, landing comfortably towards \( \frac{\pi}{2} \).
• The resultant \( 2(\tan^{-1}n) \) scales this reach by introducing a simple stretch to \( \pi \).

The inverse trigonometric operations illustrate how alignment into an angle occurs at infinity, critical for supporting the imaginary component of convergence.

Limit of a Sequence

When seeking the convergence of a sequence, achieving its limit is like catching the grand finale of an intricate dance. Think of it as defining the ultimate destination of both the real and imaginary journeys simultaneously.
In the sequence \( e^{1/n} + 2(\tan^{-1}n)i \), we've already established that

• the real part settles at 1,
• while the imaginary part spirals to \( \pi \).

Thus, as \( n \to \infty \), the sequence naturally ties itself to \( 1 + \pi i \), it confesses a destination, harmonious and fixed.
Recognizing convergence in this context serves as a testament to knowing how intricate parts, real analysis steering continuity, and inverse functions bending limits, narrative together to chart a course toward a single convergent point.