Problem 3 - Advanced Engineering Mathematics

Question

In Problems 1-4, write out the first five terms of the given sequence. $$ \left\\{1+e^{n \pi i}\right\\} $$

Step-by-Step Solution

Answer

The first five terms are 0, 2, 0, 2, 0.

Step 1: Understand the Sequence

The sequence given is \( \{1+e^{n \pi i}\} \). Here, \( n \) represents the term number, and \( e^{n \pi i} \) involves Euler's formula, where \( e^{x i} = \cos(x) + i\sin(x) \).

Step 2: Recognize the Periodicity of Exponential

For exponential terms of the form \( e^{n \pi i} \), notice that \( e^{\pi i} = -1 \) and \( e^{2\pi i} = 1 \). Therefore, these terms oscillate between \(-1\) and \(1\) based on whether \( n \) is odd or even.

Step 3: Calculate the Terms Based on Odd and Even Values of n

When \( n \) is odd, \( e^{n \pi i} = -1 \), leading to \( 1 + e^{n \pi i} = 0 \). When \( n \) is even, \( e^{n \pi i} = 1 \), leading to \( 1 + e^{n \pi i} = 2 \).

Step 4: Write the First Five Terms of the Sequence

Using the findings, calculate the first five terms: - For \( n = 1 \), \( 1 + e^{1 \pi i} = 0 \).- For \( n = 2 \), \( 1 + e^{2 \pi i} = 2 \).- For \( n = 3 \), \( 1 + e^{3 \pi i} = 0 \).- For \( n = 4 \), \( 1 + e^{4 \pi i} = 2 \).- For \( n = 5 \), \( 1 + e^{5 \pi i} = 0 \).

Key Concepts

Euler's Formula

Euler's formula is a fascinating concept in mathematics that relates exponential functions and trigonometry. It is given by the equation \( e^{ix} = \cos(x) + i\sin(x) \), where \(e\) is the base of natural logarithms, \(i\) is the imaginary unit, and \(x\) is a real number. This formula represents a complex exponential function as a point on the unit circle in the complex plane. It elegantly connects complex exponentials and trigonometric functions. No wonder it is often considered one of the most beautiful equations in mathematics.

• Key Idea: Euler's formula transforms complex exponentials into circles.
• Applications: Useful in various fields like engineering, physics, and signal processing.
• Simplifies Calculations: Helps simplify calculations involving complex numbers.

Understanding Euler's formula can greatly simplify problems that involve trigonometric identities and exponential growth, especially in the complex plane.

Exponential Functions

Exponential functions are a type of mathematical function that involves the constant \(e\), approximately equal to 2.71828. They are expressed in the form \(f(x) = e^{kx} \), where \(k\) is a constant. These functions have the unique property of having a constant rate of growth proportional to their value. This makes them ideal for modeling natural processes that grow continuously and rapidly, like populations, radioactive decay, or interest in a bank account.
In the expression \( e^{n\pi i} \), found in the sequence problem, the variable \(n\) affects the function by changing the exponent. The function cycles between values due to the imaginary unit \(i\), which highlights the periodic nature of some exponential functions.

• Rapid Growth: The value grows exponentially for real exponents.
• Complex Plane: Leads to oscillation along the unit circle for imaginary exponents.

This characteristic of exponential functions to change their behavior between growth and oscillation shows their versatility and power in multiple mathematical contexts.

Odd and Even Numbers

Odd and even numbers are a fundamental concept in mathematics. An even number is any integer that is exactly divisible by 2, while an odd number leaves a remainder of 1 when divided by 2. This simple division helps classify numbers into these two categories, providing an easy way to predict patterns and properties in sequences or sets.

• Even Numbers: Examples include ..., -4, -2, 0, 2, 4, ...
• Odd Numbers: Examples include ..., -3, -1, 1, 3, 5, ...

In the sequence problem, the distinction between odd and even numbers is key. The expression \(e^{n\pi i}\) behaves differently depending on whether \(n\) is odd or even:

• For odd \(n\), it yields \(-1\), producing \(1 + e^{n\pi i} = 0\).
• For even \(n\), it yields \(1\), producing \(1 + e^{n\pi i} = 2\).

Recognizing whether a number is odd or even can simplify the process of predicting and understanding patterns in sequences.