Problem 10 - Advanced Engineering Mathematics

Question

In Problems 5-10, 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: Identify the Sequence Terms

The given sequence is \( \{e^{1/n} + 2(\tan^{-1} n)i\} \). It consists of a real part \( e^{1/n} \) and an imaginary part \( 2(\tan^{-1} n)i \). To determine convergence, we must analyze the behavior as \( n \to \infty \) for both parts.

Step 2: Analyze the Real Part

Consider the real part of the sequence: \( e^{1/n} \). As \( n \) approaches infinity, \( \frac{1}{n} \) approaches zero. Thus, \( e^{1/n} \to e^0 = 1 \). The real part converges to 1.

Step 3: Analyze the Imaginary Part

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

Step 4: Combine Real and Imaginary Parts

Combining both parts, the sequence converges to \( 1 + \pi i \) as \( n \to \infty \). This indicates that the sequence \( \{e^{1/n} + 2(\tan^{-1} n)i\} \) converges to the complex number \( 1 + \pi i \).

Key Concepts

Real and Imaginary Parts Convergence

A complex sequence is composed of real and imaginary parts. To understand whether a complex sequence converges, it's essential to look at these parts separately. When dealing with a complex sequence like \( \{e^{1/n} + 2(\tan^{-1} n)i\} \), each part must be studied as \( n \to \infty \).

### Real Part ConvergenceThe real part here is \( e^{1/n} \). The exponential function is sensitive to changes in \( n \). As \( n \) gets larger, \( \frac{1}{n} \) becomes very small, approaching zero. Therefore, \( e^{1/n} \to e^0 = 1 \). This result indicates the real part converges to 1.

### Imaginary Part ConvergenceThe imaginary component is represented by \( 2(\tan^{-1} n)i \). In mathematics, the inverse tangent function \( \tan^{-1} n \) approaches \( \frac{\pi}{2} \) as \( n \to \infty \). Therefore, multiplying by 2, the imaginary part converges to \( \pi i \).

By understanding and analyzing each part separately, and then jointly, it can be confirmed that the sequence \( \{e^{1/n} + 2(\tan^{-1} n)i\} \) converges to a stable complex number as \( n \to \infty \). The entire sequence then converges to \( 1 + \pi i \).

Limit of Exponential Function

Understanding the behavior of exponential functions in the context of limits is crucial for analyzing sequences like \( e^{1/n} \). Exponential functions, denoted as \( e^x \), are pervasive in mathematics and sciences.

### Behavior as Argument Approaches ZeroAs \( n \to \infty \), the expression \( \frac{1}{n} \) becomes extremely small, trending towards zero. This transforms our function to \( e^{1/n} \to e^0 \). Calculating this, we find \( e^0 = 1 \).

### Key Properties

• Exponential Growth/Decay: Exponential functions display rapid growth or decay. With base \( e \) raised to a negative or very small power, the function converges towards moderate values.
• Continuity: Since the exponential function is continuous, as the input approaches zero (from positive values), the output seamlessly approaches the limit value.

Thus, for sequences involving exponential terms, the limit is dictated by their behavior as their exponent trends towards zero or other small values.

Limit of Inverse Trigonometric Function

Inverse trigonometric functions, like \( \tan^{-1} \), have specific limit behaviors essential to calculus. They are instrumental in analyzing sequences involving complex numbers.

### Behavior of \( \tan^{-1} n \) as \( n \to \infty \)For purposes of convergence analysis, it’s important to know that as \( n \to \infty \), \( \tan^{-1} n \) heads towards its horizontal asymptote of \( \frac{\pi}{2} \). Consequently, the arc tangent function slowly increases, approaching but never quite reaching \( \frac{\pi}{2} \). This behavior contributes significantly to the imaginary part of the sequence.

### Practical Considerations

• Range of Output: The function outputs values between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\), making it bounded.
• Smooth Convergence: Despite its slower approach compared to linear functions, its convergence is smooth and predictable.

Thus, multiplying \( \tan^{-1} \) by a constant, such as 2, will keep the limit behavior in check by simply scaling the convergence result but not altering the fact that it converges to a limit like \( \pi \) in this context.