Problem 39

Question

Does the sequence $\left\\{i^{1 / n}\right\\}$, where $i^{1 / n}$ denotes the principal $n$ th root of $i$, converge?

Step-by-Step Solution

Verified

Answer

The sequence $\{i^{1/n}\}$ converges to 1.

1Step 1: Understanding the Principal nth Root

The principal nth root of a complex number like the imaginary unit $i$ can be expressed in polar form. Remember that $i = e^{i\frac{\pi}{2}}$ in polar form. For the nth root, this can be written as $i^{1/n} = e^{i(\frac{\pi}{2n} + \frac{2k\pi}{n})}$, where $k$ is an integer.

2Step 2: Investigating the Argument

The angle (or argument) of the nth root for $i$ is $\theta_n = \frac{\pi}{2n}$ plus any rotation by a full circle, $\frac{2k\pi}{n}$. Focusing on $\frac{\pi}{2n}$, as $n$ tends to infinity, this angle approaches zero.

3Step 3: Converting to Rectangular Form

When $i^{1/n} = e^{i\theta_n}$ and $\theta_n$ approaches zero, the expression approaches 1 in rectangular form because $e^{i\theta} = \cos(\theta) + i\sin(\theta)$, and when $\theta$ is small, $\cos(\theta)\approx 1$ and $\sin(\theta)\approx \theta$. Thus, the roots approach 1 as $n$ becomes large.

4Step 4: Conclusion of Convergence

With each $i^{1/n}$ getting closer to 1 as $n$ increases, the sequence $\{i^{1/n}\}$ converges to 1. Since both the real and imaginary parts become closer to the real number 1, this confirms convergence.

Key Concepts

Principal nth RootPolar FormConvergence

Principal nth Root

The principal nth root is a fundamental concept in complex number theory, crucial when dealing with complex numbers like the imaginary unit $i$. Essentially, it refers to finding one particular root among the multiple possibilities that exist for any number in the complex plane.

For the complex number $i$, we can apply the formula for the nth root in polar form. Polar form is a way of expressing a complex number using a magnitude (or modulus) and an angle (or argument), instead of the usual rectangular form $(a + bi)$. The number $i$ itself can be written in polar form as $e^{i\frac{\pi}{2}}$.

Root formula: $i^{1/n} = e^{i(\frac{\pi}{2n} + \frac{2k\pi}{n})}$
The term $\frac{\pi}{2n}$ represents the principal root, while $\frac{2k\pi}{n}$ corresponds to all possible nth roots.
$k$ is an integer, indicating the different rotations or directions.

Understanding the principal nth root is essential when analyzing how sequences of complex numbers function and behave as the index $n$ increases.

Polar Form

Polar form is a key method for representing complex numbers, especially useful when dealing with powers and roots. Instead of using coordinates, polar form describes a number's magnitude and direction from the origin.
Polar coordinates are usually given as $r(\cos(\theta) + i\sin(\theta))$ or $re^{i\theta}$, where:

$r$ is the modulus, representing the distance from the origin. For $i$, this is 1, since $i = (0, 1)$ on the complex plane.
$\theta$ is the argument, an angle measured from the positive real axis. For $i$, this angle is $\frac{\pi}{2}$.

Using polar form simplifies the process of finding roots and powers because it allows a straightforward multiplication and addition of angles.

When we convert $i^{1/n}$ to polar form, the angle part $\frac{\pi}{2n}$ gets smaller as $n$ increases, leading the sequence towards convergence. This shows the elegance and utility of polar form in simplifying calculations and understanding the behavior of complex sequences.

Convergence

Convergence in the context of complex sequences involves analyzing how a sequence behaves as its terms progress towards a specific value. In this case, for $\{i^{1/n}\}$, convergence means examining if the sequence approaches a single point as $n$ becomes very large.

From our understanding of the principal nth root and polar form, we see:

For $i^{1/n} = e^{i\theta_n}$, we look at the angle $\theta_n = \frac{\pi}{2n}$.
As $n$ increases, $\theta_n$ shrinks and approaches zero.
In the limit, $e^{i\theta_n}$ approaches $e^0 = 1$.

So, as $n$ grows, the sequence $\{i^{1/n}\}$ converges to 1 in the complex plane. It showcases how both the real and imaginary components align closer to 1, confirming the sequence's tendency towards a real limit. Convergence is a crucial property in mathematical analysis, helping us determine stability and limits within complex number sequences.

Problem 39

Problem 40

Other exercises in this chapter

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Problem 41

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