Problem 2 - Thomas Calculus

Question

Use the Comparison Test to determine if each series converges or diverges. \begin{equation}\sum_{n=1}^{\infty} \frac{n-1}{n^{4}+2}\end{equation}

Step-by-Step Solution

Answer

The series converges by the Comparison Test.

Step 1: Identify the Series to Test

We are given the series \( \sum_{n=1}^{\infty} \frac{n-1}{n^4+2} \). Our task is to determine whether this series converges or diverges using the Comparison Test.

Step 2: Simplify the General Term

The general term of the series is \( \frac{n-1}{n^4+2} \). For large \( n \), \( n-1 \approx n \) and \( n^4+2 \approx n^4 \). This simplifies to \( \frac{n}{n^4} = \frac{1}{n^3} \). We will use this to compare the original series with a series we know converges or diverges.

Step 3: Choose a Comparison Series

We choose the series \( \sum_{n=1}^{\infty} \frac{1}{n^3} \) as our comparison series. This is a \( p \)-series with \( p = 3 \), and since \( p > 1 \), this series converges.

Step 4: Apply the Comparison Test

To use the Comparison Test, we compare \( \frac{n-1}{n^4+2} \) with \( \frac{1}{n^3} \). For \( n \geq 2 \), we have \( n-1 < n \) and \( n^4+2 > n^4 \), so \( \frac{n-1}{n^4+2} < \frac{n}{n^4} = \frac{1}{n^3} \). Since \( \frac{1}{n^3} \) converges, by the Comparison Test, \( \sum_{n=1}^{\infty} \frac{n-1}{n^4+2} \) also converges.

Step 5: Conclude the Result

We have shown that \( \frac{n-1}{n^4+2} < \frac{1}{n^3} \) for all \( n \geq 2 \), and since the series \( \sum_{n=1}^{\infty} \frac{1}{n^3} \) converges, the original series \( \sum_{n=1}^{\infty} \frac{n-1}{n^4+2} \) must also converge by the Comparison Test.

Key Concepts

Convergence and Divergence of Series

When learning about the convergence and divergence of series, it is important to understand how a series behaves as the number of terms grows infinitely large. A series is an infinite sum, represented symbolically as \(\sum_{n=1}^{\infty} a_n\), where \(a_n\) are the terms of the series. The main goal is to determine whether this sum approaches a finite number (convergence) or not (divergence).
To analyze this, several tests are available. One of the most commonly used methods is the Comparison Test. It compares a series whose convergence status is unknown with another series whose status is already known. If the series in question is smaller than or of a similar form to a known convergent series, the original series is also convergent. Conversely, if it is larger than a known divergent series, it diverges as well.

• Convergence: The series approaches a finite limit.
• Divergence: The series grows without bound or oscillates.

Recognizing convergence or divergence helps in understanding the behavior and potential application of mathematical sequences in calculus.

P-Series

In sequences and series, a p-series is a useful and significant type of series, which has the form \(\sum_{n=1}^{\infty} \frac{1}{n^p}\). It is one of the essential building blocks for understanding series convergence. Whether a p-series converges or diverges depends solely on the power \(p\):

• If \(p > 1\), the p-series converges.
• If \(p \leq 1\), the p-series diverges.

For instance, in a problem where you are using the Comparison Test, you can compare your series to a p-series to determine convergence, as demonstrated in our original series exercise. By simplifying the given series \(\sum_{n=1}^{\infty} \frac{n-1}{n^4+2} \) to the p-series model \(\frac{1}{n^3}\), we conclude it converges because \(p = 3\) which is greater than 1.
P-series makes for an excellent comparison due to its straightforward convergence criteria and clear cutoff between converging and diverging scenarios.

Sequences and Series

Understanding the difference between sequences and series is fundamental to grasping their individual functions in calculus. A sequence is simply a list of numbers following a certain pattern, expressed as \( \{a_n\} \). Depending on the formula, sequences can be finite or infinite.
On the other hand, a series is the sum of a sequence of numbers. When we add up elements from a sequence, we get a series. The infinite nature of these sequences can create unique challenges in terms of computation and prediction of series behavior.

• Sequences: Ordered list of numbers such as \(a_1, a_2, a_3, \ldots\).
• Series: The sum of a sequence, \(\sum_{n=1}^{\infty} a_n\).

Series are prolific in applications ranging from mathematical theorems to solving real-world physics problems. Learning how to analyze series through different tests, like the Comparison Test, enables us to conclude whether a series converges or diverges, thus broadening our mathematical understanding.