Alternating Series Convergence: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving deep into the fascinating world of alternating series and figuring out whether a specific one converges or, sadly, diverges. The series we're tackling is $\sum_{n=1}^{\infty}(-1)^{n+1}\left(\frac{n}{22}\right)^n$. It might look a bit intimidating at first glance, but trust me, we'll break it down step by step and make it super understandable. We'll use the tools of mathematical analysis to determine its fate.

Unveiling the Magnitude: Defining $u_n$

First things first, let's understand what we're working with. In the context of an alternating series, we have this general form: $\sum_{n=1}^{\infty}(-1)^{n+1}a_n$ or $\sum_{n=1}^{\infty}(-1)^{n}a_n$. Here, the $(-1)^{n+1}$ (or $(-1)^{n}$) is what gives us the alternating behavior. The $a_n$ part is what we're really interested in, as it defines the magnitude or absolute value of the terms. In our specific series, $\sum_{n=1}^{\infty}(-1)^{n+1}\left(\frac{n}{22}\right)^n$, we can represent the magnitude of the terms, which we denote as $u_n$, as $u_n = \left|\left(\frac{n}{22}\right)^n\right|$. Since $n$ is always positive (because it's the index of summation), $u_n$ simplifies to $u_n = \left(\frac{n}{22}\right)^n$. So, $u_n$ represents the absolute value of each term in the series without the alternating $(-1)^{n+1}$ part. It is important to identify $u_n$ for our subsequent tests.

Now, let's break down why defining $u_n$ is crucial. Imagine you're trying to figure out if a series converges. Alternating series have a special set of rules, but the core idea is to understand the behavior of the terms without the alternating sign. This is where $u_n$ comes in. It isolates the magnitude, letting us focus on whether those magnitudes get smaller and smaller (and approach zero) as $n$ goes to infinity. The alternating sign tells us how those magnitudes are combined, but the $u_n$ part is the real driver of convergence or divergence. We need to investigate if the terms approach zero. The analysis of $u_n$ allows us to apply the alternating series test and other convergence tests more effectively. Identifying $u_n$ is a critical early step. It's the foundation for understanding the behavior of the series terms. Without a clear definition of $u_n$, we'd be stumbling in the dark, unable to apply the necessary tests and draw accurate conclusions about the series' convergence.

The Ratio Test: Our Weapon of Choice

Alright, folks, with $u_n$ defined, we need a way to determine whether the series $\sum_{n=1}^{\infty}(-1)^{n+1}\left(\frac{n}{22}\right)^n$ converges or diverges. Because our $u_n$ has a power of $n$, this screams for a test that deals well with exponents. The Ratio Test is perfect for this task. It's especially useful when dealing with terms that have factorials or, in our case, something raised to the power of $n$.

The Ratio Test says we need to calculate the limit of the absolute value of the ratio of consecutive terms. Specifically, we're looking at: $\lim_n\to\infty} \left|\frac{u_{n+1}}{u_n}\right|$ Where $u_n = \left(\frac{n}{22}\right)^n$. First, let's determine $u_{n+1}$ $u_{n+1 = \left(\fracn+1}{22}\right)^{n+1}$ Now, let's form the ratio $\frac{u_{n+1}}{u_n}$ $\frac{u_{n+1}u_n} = \frac{\left(\frac{n+1}{22}\right)^{{n+1}}{\left(\frac{n}{22}\right)}n} = \frac{(n+1)^{n+1}}{22{n+1}} \cdot \frac{22^n}{nn} = \frac{(n+1)^{n+1}}{nn}\cdot\frac{1}{22}$ Simplifying a bit further, we get $\frac{u_{n+1}u_n} = \frac{(n+1)^n(n+1)}{nn}\cdot\frac{1}{22} = \left(1 + \frac{1}{n}\right)^n\cdot\frac{n+1}{22}$ Now, we need to take the limit as $n$ approaches infinity $\lim_{n\to\infty \left|\fracu_{n+1}}{u_n}\right| = \lim_{n\to\infty} \left(1 + \frac{1}{n}\right)^n\cdot\frac{n+1}{22}$ The limit $\lim_{n\to\infty} \left(1 + \frac{1}{n}\right)^n$ is a classic. It equals e (Euler's number, approximately 2.71828). However, the $\frac{n+1}{22}$ part goes to infinity as $n$ goes to infinity. So, we have $\lim_{n\to\infty \left|\frac{u_{n+1}}{u_n}\right| = e \cdot \lim_{n\to\infty} \frac{n+1}{22} = \infty$

Making the Call: Convergence or Divergence?

So, after all that work, what does this tell us? The Ratio Test provides a definitive answer:

• If $\lim_{n\to\infty} \left|\frac{u_{n+1}}{u_n}\right| < 1$, the series converges.
• If $\lim_{n\to\infty} \left|\frac{u_{n+1}}{u_n}\right| > 1$ or $\infty$, the series diverges.
• If $\lim_{n\to\infty} \left|\frac{u_{n+1}}{u_n}\right| = 1$, the test is inconclusive.

In our case, the limit is infinity. Therefore, according to the Ratio Test, the series $\sum_{n=1}^{\infty}(-1)^{n+1}\left(\frac{n}{22}\right)^n$ diverges. That's the final verdict! The terms of the series don't approach zero fast enough, and the series goes off to infinity. This conclusion is vital for understanding the behavior of this particular alternating series, which is a key concept in calculus and mathematical analysis. The divergence means the series doesn't settle down to a specific value; it keeps growing without bound. Understanding why the series diverges is a fundamental skill in mastering calculus and related fields. It means that, as we add more and more terms, the sum doesn't get closer to a finite number; instead, it oscillates with increasing magnitude, failing to converge to a single value.

In Summary

We started with an alternating series and defined $u_n$ to represent the magnitude of its terms. Then, we used the Ratio Test to evaluate the limit of the ratio of consecutive terms. Because the limit was infinite, we concluded that the series diverges. Remember, understanding these tests and how to apply them is a crucial skill for any math student. Keep practicing, and you'll become a convergence/divergence expert in no time! Keep exploring and enjoy the world of math!