Problem 41 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Determine whether the series converges or diverges. In this set of problems knowledge of all the convergence tests from the chapter is assumed. \(\sum_{k=1}^{\infty}\left(\frac{k^{2}-3 k}{5 k^{2+1}}\right)^{k}\)

Step-by-Step Solution

Answer

The series \(\sum_{k=1}^{\infty}\left(\frac{k^{2}-3 k}{5 k^{3}}\right)^{k}\) converges.

Step 1: Simplify the expression

The first step is to simplify the function \(f(k) = \left(\frac{k^{2}-3 k}{5k^{3}}\right)^k\). This can be rewritten as \(f(k) = \left(\frac{1 - \frac{3}{k}}{5k}\right)^k\). Simplifying the expression further: \(f(k) = \left(\frac{1}{k} - \frac{3}{k^2}\right)^k/5^k\).

Step 2: Apply the Root Test

As the order of the root is \(k\), the Root Test is the most suitable test for analyzing the convergence of the series. The root test is given by the formula \(\lim_{k \to \infty}|a_k|^{1/k}\), where \(a_k\) is the general term in the series.

Step 3: Perform the Root Test

The Root Test says that if \(\lim_{k \to \infty}\sqrt[k]{|a_k|}\) exists and if it is less than 1, then the series \(\sum a_k\) is convergent. Applying the Root test to our series gives: \(\lim_{k \to \infty}\sqrt[k]{|f(k)|}\) = \(\lim_{k \to \infty} \frac{1}{k} - 3/k^2\)/\(\sqrt[k]{5}\). As the limit of \(f(k)\) when \(k\) approaches infinity is zero, and \(\sqrt[k]{5}\) goes to 1, the series converges.

Step 4: Result

Based on the Root Test, since the composition of the limit reached is smaller than 1, the original series \(\sum_{k=1}^{\infty}\left(\frac{k^{2}-3 k}{5 k^{3}}\right)^{k}\) converges.

Key Concepts

Root Test

In the context of determining whether an infinite series converges or diverges, the Root Test is a powerful tool. It's particularly useful when terms of the sequence include powers or roots. The Root Test examines the behavior of the sequence of terms \(a_k\) as \(k \ o \infty\).

The Root Test formula is:

• Calculate \[ \lim_{{k \to \infty}} \sqrt[k]{|a_k|} \]
• If this limit is less than 1, the series converges absolutely.
• If the limit is greater than 1, or the limit is infinite, the series diverges.
• If the limit equals 1, the test is inconclusive, and another method is needed.

Understanding this helps greatly in penetrating problems involving sequences with powers, ensuring the right method is applied. When the series \[ \sum_{k=1}^{\infty} \left(\frac{k^2-3k}{5k^3}\right)^k \] was tested with the Root Test, it converged since the limit turned out to be less than 1.

Infinite Series

An infinite series is a sequence of summed terms that extend indefinitely. It is written as \( \sum_{k=1}^{\infty} a_k \), where \(a_k\) represents the terms of the series. The series is called convergent if the series approaches a specific finite value as more terms are added; otherwise, it's called divergent.

Infinite series are used throughout mathematics and many sciences for modeling and solving real-world problems. For instance, they help in calculating areas under curves, solving differential equations, and in various computational algorithms.

• Convergent series approach a finite number.
• Divergent series do not settle into a finite limit as more terms are added.

Series like \[ \sum_{k=1}^{\infty} \left(\frac{k^2 - 3k}{5k^3}\right)^k \] require convergence tests to determine their behavior, bridging concepts of calculus with algebra.

Convergence Tests

Convergence tests are various methods used to determine whether an infinite series converges or diverges. These tests provide systematic approaches in assessing series accuracy, especially when the series doesn't fit simple forms.

Common tests include:

• Comparison Test: Compares a given series with a known convergent or divergent series.
• Ratio Test: Utilizes the ratio of consecutive terms to establish convergence.
• Root Test: Evaluates the \( k^{th} \) root of absolute terms for convergence.
• Integral Test: Compares the series with an improper integral to infer convergence.

Each test is suited for different forms of series, making it critical to choose the most efficient method. The Root Test stands out when dealing with exponential terms like those encountered in \[ \sum_{k=1}^{\infty} \left(\frac{k^2-3k}{5k^3}\right)^k \], simplifying the analysis process and leading to decisive conclusions about convergence.