Problem 12 - A Course in Calculus and Real Analysis

Question

For \(k \in \mathbb{N}\), let \(a_{2 k-1}:=4^{k-1} / 9^{k-1}\) and \(a_{2 k}:=4^{k-1} / 9^{k} .\) Show that \(\left|a_{2 k} / a_{2 k-1}\right|=\frac{1}{9}\) and \(\left|a_{2 k+1} / a_{2 k}\right|=4\) for all \(k \in \mathbb{N}\) and so the Ratio Test for the convergence of \(\sum_{k=1}^{\infty} a_{k}\) is inconclusive. Prove that \(\left|a_{k}\right|^{1 / k} \rightarrow \frac{2}{3}\) as \(k \rightarrow \infty\) and use the Root Test to conclude that \(\sum_{k=1}^{\infty} a_{k}\) is convergent.

Step-by-Step Solution

Answer

We found the absolute values of the ratios of consecutive terms and showed that \(\left|a_{2k}/a_{2k-1}\right| = \frac{1}{9}\) and \(\left|a_{2k+1}/a_{2k}\right| = 4\) for all \(k \in \mathbb{N}\). Since these values aren't constant, the Ratio Test is inconclusive. Computing the limit for the Root Test, we found that \(\lim_{k \rightarrow \infty} \left|a_k\right|^{1/k} = \frac{2}{3}\), which is less than 1. Therefore, by the Root Test, the series \(\sum_{k=1}^\infty a_k\) converges.

Step 1: Find the ratios of consecutive terms for both sequences

To find the Ratio Test ratios, we want to compute \(\left|a_{2k}/a_{2k-1}\right|\) and \(\left|a_{2k+1}/a_{2k}\right|\) for the given sequences. For \(\left|a_{2k}/a_{2k-1}\right|\), we have: \(\left|\frac{4^{k-1}/9^k}{4^{k-1}/9^{k-1}}\right| = \left|\frac{4^{k-1}}{9^{k}}\right|\times \left|\frac{9^{k-1}}{4^{k-1}}\right| = |9^{-1}| = \frac{1}{9}\) For \(\left|a_{2k+1}/a_{2k}\right|\), we have: \(\left|\frac{4^k/9^k}{4^{k-1}/9^k}\right| = \left|\frac{4^k}{4^{k-1}}\right| = 4\) Thus, we have shown that \(\left|a_{2k}/a_{2k-1}\right| = \frac{1}{9}\) and \(\left|a_{2k+1}/a_{2k}\right| = 4\) for all \(k \in \mathbb{N}\).

Step 2: The Ratio Test is inconclusive

Since the absolute values of the ratios of consecutive terms aren't constant, the Ratio Test cannot determine the convergence of the series.

Step 3: Compute the limit for the Root Test

To apply the Root Test, we need to find the limit of \(\left|a_k\right|^{1/k}\) as \(k \rightarrow \infty\). Since the series has different expressions for the even and odd terms, we can compute the separate limits for both even and odd k. For odd k, \(k = 2n - 1\) for some \(n \in \mathbb{N}\): \(\lim_{n\to\infty}\left(\frac{4^{n-1}}{9^{n-1}}\right)^{1/(2n-1)} = \lim_{n\to\infty}\left(\frac{2^{2(n-1)}}{3^{2(n-1)}}\right)^{1/(2n-1)} = \lim_{n\to\infty}\left(\frac{2}{3}\right)^{2/(2n-1)} = \left(\frac{2}{3}\right)^2\) For even k, \(k = 2n\) for some \(n \in \mathbb{N}\): \(\lim_{n\to\infty}\left(\frac{4^{n-1}}{9^n}\right)^{1/(2n)} = \lim_{n\to\infty}\left(\frac{2^{2(n-1)}}{3^{2n}}\right)^{1/(2n)} = \lim_{n\to\infty}\left(\frac{2}{3}\right)^{2n/(2n)} = \frac{2}{3}\) Since both limits are equal, we conclude that \(\lim_{k\to\infty} \left|a_k\right|^{1/k} = \frac{2}{3}\).

Step 4: Apply the Root Test

We have shown that \(\lim_{k\to\infty} \left|a_k\right|^{1/k} = \frac{2}{3}\), which is less than 1. By the Root Test, series \(\sum_{k=1}^\infty a_k\) converges.

Key Concepts

Ratio Test

The Ratio Test is a popular method in calculus used to determine whether a series converges or diverges. You calculate the limit of the absolute value of the ratio of consecutive terms. Specifically, for a series \( \sum a_k \), you find \( \lim_{k \to \infty} \left|\frac{a_{k+1}}{a_k}\right| \).
This can tell us:

• If the limit is less than 1, the series converges absolutely.
• If the limit is greater than 1, the series diverges.
• If the limit equals 1, the test is inconclusive.

In this exercise, we examined two ratio expressions \( \left|a_{2k} / a_{2k-1}\right| \) and \( \left|a_{2k+1} / a_{2k}\right| \). These do not yield a consistent result. Therefore, for the given series, the Ratio Test neither proves convergence nor divergence. We called the test 'inconclusive,' meaning other methods, like the Root Test, must be considered.

Root Test

The Root Test, also known as the Radical Test, helps analyze the convergence of a series by examining the limit of the \( k \)-th root of the absolute term itself. Here we find \( \lim_{k \to \infty} (|a_k|)^{1/k} \).
This tells us:

• If this limit is less than 1, the series converges absolutely.
• If greater than 1, the series diverges.
• If it is equal to 1, the test is inconclusive.

In this situation, after calculating separately for even and odd indexed terms, we found that the limit becomes \( \frac{2}{3} \) as \( k \) grows infinitely large. Since \( \frac{2}{3} < 1 \), the Root Test strongly indicates the series converges. This conclusion effectively complements the inconclusive result from the Ratio Test.

Sequences and Series

In mathematics, sequences and series are foundational concepts. A sequence is simply a list of numbers ordered in a specific, logical way. Example: \( a_1, a_2, a_3, \ldots \). Each number in the list corresponds to a natural number index.
A series is the sum of the terms of a sequence. If a sequence \( a_k \) is finite and you add them up, you have a finite series. However, in calculus, we most often work with infinite series, which is the ongoing summation of an infinite sequence \( \sum_{k=1}^{if} a_k \). Understanding when these infinite series converge is crucial. Identifying whether a series converges means determining if the sum approaches a specific number as more of its terms are included.

Convergence of Series

The convergence of series tells us if the sum of series results in a finite number. A series \( \sum_{k=1}^{if} a_k \) is said to converge if the sequence of partial sums \( S_n = a_1 + a_2 + \ldots + a_n \) converges to a limit \( S \) as \( n \to \infty \).
There are several methods and tests to determine convergence:

• Ratio Test is useful for series with factorials or constant powers.
• Root Test is effective for series with exponential growth or roots.
• Other tests include comparison test, integral test, and more.

For a series, it is essential first to examine the behavior of its terms as \( k \to \infty \). In our problem, the Root Test enabled us to establish that our series converges as the limit of the \( k \)-th root of the absolute value of terms \( \left|a_k\right|^{1/k} \) approaches less than one, offering a definitive convergence conclusion.