Problem 1 - Thomas Calculus

Question

Use the Comparison Test to determine if each series converges or diverges. \begin{equation}\sum_{n=1}^{\infty} \frac{1}{n^{2}+30}\end{equation}

Step-by-Step Solution

Answer

The series converges by the Comparison Test.

Step 1: Recognize the Series

The series given is \( \sum_{n=1}^{\infty} \frac{1}{n^{2}+30} \). We need to use the Comparison Test to analyze its convergence or divergence.

Step 2: Choose a Comparison Series

For the comparison series, we choose the well-known convergent p-series \( \sum_{n=1}^{\infty} \frac{1}{n^2} \), which is known to converge because \( p = 2 > 1 \).

Step 3: Establish Inequality for Comparison

We aim to compare \( \frac{1}{n^{2}+30} \) with \( \frac{1}{n^2} \). Since \( n^2 < n^2 + 30 \) for all natural numbers \( n \), it follows that \( \frac{1}{n^2 + 30} < \frac{1}{n^2} \).

Step 4: Apply the Comparison Test

According to the Comparison Test, if \( 0 \leq a_n \leq b_n \) for all \( n \) and \( \sum b_n \) converges, then \( \sum a_n \) also converges. Here, with \( a_n = \frac{1}{n^2 + 30} \) and \( b_n = \frac{1}{n^2} \), since \( \sum_{n=1}^{\infty} \frac{1}{n^2} \) converges, so does \( \sum_{n=1}^{\infty} \frac{1}{n^2 + 30} \).

Key Concepts

p-series

Understanding p-series is crucial to analyzing many mathematical series. A p-series is a specific kind of infinite series expressed in the form \( \sum_{n=1}^{\infty} \frac{1}{n^p} \), where \( p \) is a positive real number.

The convergence of a p-series heavily relies on the value of \( p \):

• If \( p > 1 \), the series converges. This means the sum of the infinite series results in a finite number.
• If \( p \leq 1 \), the series diverges, implying the sum approaches infinity.

In the given solution, we used the p-series \( \sum_{n=1}^{\infty} \frac{1}{n^2} \) for comparison. Here, \( p = 2 \), which is greater than 1, thereby guaranteeing convergence. This foundational concept in calculus forms the basis for utilizing the Comparison Test effectively.

convergence

Convergence in series analysis refers to the behavior of the sums of the terms as more terms are added. Specifically, a series converges if the sequence of its partial sums tends to a finite limit.

To determine convergence, mathematicians often employ tools like the Comparison Test. If the chosen comparison series is known to converge and it bounds the original series from above, the original series is also said to converge. In this exercise, the series \( \sum_{n=1}^{\infty} \frac{1}{n^2+30} \) was compared to the convergent p-series \( \sum_{n=1}^{\infty} \frac{1}{n^2} \).

By establishing that the terms in the original series are always less than those in the convergent series, the Comparison Test confirms that the series under study also converges. Convergence is a powerful tool that helps to understand not just the behavior but also the outcomes of infinite processes or signals.

inequality comparison

The concept of inequality comparison is pivotal when applying the Comparison Test to determine convergence or divergence of series. The essence of this technique lies in setting up an inequality between the terms of the series in question and a known series.

In this exercise, it was established that \( n^2 < n^2 + 30 \) for all natural numbers \( n \). From this inequality, it follows that \( \frac{1}{n^2+30} < \frac{1}{n^2} \).

With this relationship in place, the Comparison Test states that if \( 0 \leq a_n \leq b_n \) for each \( n \), and the series \( \sum b_n \) converges, then \( \sum a_n \) converges too. Here, \( a_n = \frac{1}{n^2+30} \) and \( b_n = \frac{1}{n^2} \), with the latter series already known to converge.

Therefore, through inequality comparison, the test provides a means to extrapolate the convergence behavior from a simpler or already established series to a more complex one, greatly simplifying the analysis process.