Problem 9 - Advanced Engineering Mathematics

Question

Determine whether the given sequence converges or diverges. \(\left\\{\frac{n+i^{n}}{\sqrt{n}}\right\\}\)

Step-by-Step Solution

Answer

The sequence diverges.

Step 1: Simplify the Sequence

First, examine the sequence \(a_n = \frac{n+i^n}{\sqrt{n}}\). The imaginary unit \(i\) cycles through four values: \(i, -1, -i, 1\). This means \(i^n\) can be one of these values depending on \(n\mod 4\). Rewrite the sequence considering this behavior: \(a_n = \frac{n}{\sqrt{n}} + \frac{i^n}{\sqrt{n}}\). Simplify the expression to \(a_n = \sqrt{n} + \frac{i^n}{\sqrt{n}}\).

Step 2: Analyze the Contribution of Each Term

Examine the term \(\sqrt{n}\). As \(n\) increases, \(\sqrt{n}\) goes to infinity. Next, consider \(\frac{i^n}{\sqrt{n}}\). The value of \(i^n\) is bounded (since it only takes on the values \(i, -1, -i,\) or \(1\)), and \(\sqrt{n}\) grows without bound, thus \(\frac{i^n}{\sqrt{n}}\) approaches zero as \(n\) becomes very large.

Step 3: Evaluate the Dominant Term

Focus on the dominant term in \(a_n = \sqrt{n} + \frac{i^n}{\sqrt{n}}\). The term \(\sqrt{n}\) dominates because it grows infinitely larger, while \(\frac{i^n}{\sqrt{n}}\) approaches zero and doesn't affect the overall tendency towards infinity.

Step 4: Conclusion on Convergence or Divergence

Since \(\sqrt{n}\) grows to infinity and is the dominant term, \(a_n\) also grows to infinity. Therefore, the sequence \(\left\{a_n\right\} = \left\{\frac{n+i^n}{\sqrt{n}}\right\}\) diverges.

Key Concepts

Complex Numbers

Complex numbers are an extension of the real numbers that include an imaginary unit, denoted as \(i\), where \(i^2 = -1\). This provides a way to express numbers that cannot be described on the standard real number line.
A complex number consists of two parts: a real part and an imaginary part. It can be expressed in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part.

• Real Part (\(a\)): This is the portion that you would find on the real number line.
• Imaginary Part (\(bi\)): This is the component that involves \(i\), allowing representation of extra-dimensional values not covered by real numbers alone.

Complex numbers are fundamental in many fields, including mathematics, physics, and engineering, because they allow for the solution of equations that have no real solutions.
In the given sequence, \(i^n\) is a series of transformations which cycle through four possible values: \(i, -1, -i,\) and \(1\) based on the modulus perceiving the value of \(n\). This cyclical nature is important to recognize to understand behaviors and transformations in sequences.

Sequence Divergence

Sequence divergence refers to the behavior of a sequence where the terms do not approach a single finite limit as the sequence progresses. Instead, the terms might increase to infinity, decrease to negative infinity, or oscillate without ever settling at any particular value.
To determine if a sequence diverges:

• Observe the behavior of its terms as \(n\) becomes very large.
• If terms do not settle but instead grow indefinitely, the sequence is divergent.
• The term-by-term behavior can provide insight into the dominant term causing divergence.

In our case, the sequence \(a_n = \sqrt{n} + \frac{i^n}{\sqrt{n}}\) diverges as \(n\) approaches infinity. This divergence occurs because the \(\sqrt{n}\) term grows without bound, overshadowing the bounded nature of \(\frac{i^n}{\sqrt{n}}\), which approaches zero. The presence of the infinite term \(\sqrt{n}\) dictates the sequence's overall tendency towards infinity, leading to divergence.

Limits of Sequences

The limit of a sequence refers to the value that the terms of a sequence approach as \(n\) approaches infinity. If there is a specific value that the terms are getting closer to, we say the sequence converges to that limit.
For a sequence \(\{a_n\}\) to converge to a limit \(L\):

• As \(n\) becomes very large, \(a_n\) becomes arbitrarily close to \(L\).
• Mathematically, this is represented as \(\lim_{{n \to \infty}} a_n = L\).
• If no such \(L\) exists, the sequence does not converge and is considered divergent.

In addressing the given exercise, understanding limits helps in recognizing why the sequence diverges. The term \(\sqrt{n}\) steadily becomes larger, pulling the sequence towards infinity instead of a fixed limit. As such, the sequence \(\frac{n+i^n}{\sqrt{n}}\) does not reach a constant value and thus diverges. This concept is crucial for interpreting and analyzing sequences, highlighting the role limits play in sequence convergence and divergence.