Problem 1 - Calculus Single Variable

Question

In each of Exercises \(1-16,\) use the Comparison Test for Convergence to show that the given series converges. State the series that you use for comparison and the reason for its convergence. $$ \sum_{n=1}^{\infty} \frac{n}{n^{3}+1} $$

Step-by-Step Solution

Answer

The series converges by the Comparison Test with \( \sum \frac{1}{n^2} \).

Step 1: Identify the General Term

The general term of the given series is \( a_n = \frac{n}{n^3 + 1} \). To apply the Comparison Test, we need to compare this with a known convergent or divergent series.

Step 2: Simplify the General Term for Comparison

Notice that as \( n \) becomes very large, \( n^3 + 1 \approx n^3 \). Therefore, the general term \( a_n = \frac{n}{n^3 + 1} \) can be approximated by \( \frac{n}{n^3} = \frac{1}{n^2} \).

Step 3: Choose a Comparison Series

We choose the series \( \sum_{n=1}^{\infty} \frac{1}{n^2} \) for comparison because it is a known convergent p-series, with \( p = 2 > 1 \). According to the p-series test, \( \sum_{n=1}^{\infty} \frac{1}{n^2} \) converges.

Step 4: Apply the Comparison Test

For sufficiently large \( n \), we have \( 0 \leq \frac{n}{n^3 + 1} < \frac{1}{n^2} \). By the Comparison Test, since \( \sum_{n=1}^{\infty} \frac{1}{n^2} \) converges and \( \frac{n}{n^3 + 1} \) is less than the corresponding term in this convergent series, the original series \( \sum_{n=1}^{\infty} \frac{n}{n^3 + 1} \) also converges.

Key Concepts

p-series

A p-series is a specific type of infinite series characterized by the formula \( \sum_{n=1}^{\infty} \frac{1}{n^p} \), where \( p \) is a positive constant. This series is known for its clear rules about convergence based on the value of \( p \).

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

For example, the series \( \sum_{n=1}^{\infty} \frac{1}{n^2} \) is a p-series where \( p = 2 \). Since \( p = 2 > 1 \), this series converges. P-series are incredibly useful in mathematical analysis due to these straightforward convergence criteria.This property makes p-series an essential tool in determining the behavior of more complicated series by comparison, as their convergence properties are well established and can serve as benchmarks.

series convergence

Series convergence is the study of whether an infinite sum of terms results in a finite number, or if it "diverges", continuing to grow indefinitely. The concept of convergence is critical in understanding whether a series can be summed to a particular value, which has implications in various fields of mathematics and science.Two classic tests to determine convergence include:

• The Comparison Test: This test involves comparing the series in question with another series whose convergence properties are already known.
• The Limit Comparison Test: This is a slight variation where the limit of the ratio of the nth terms is considered.

Understanding convergence helps identify when calculations involving infinite series are meaningful. The goal is to ensure sums like \( \sum_{n=1}^{\infty} \frac{1}{n^2} \) yield a finite result.

mathematical series

A mathematical series is the sum of the terms of a sequence. Typically represented as \( \sum_{n=1}^{\infty} a_n \), where \( a_n \) denotes the nth term of the series. Series are central in mathematics, modeling everything from simple arithmetic sequences to complex functions.Series can be:

• Finite: Where the sum involves a limited number of terms.
• Infinite: Where the series continues indefinitely.

Infinite series, like the one described in the original problem, are of particular interest. Whether such a series converges or diverges depends largely on the behavior of its terms as \( n \) becomes very large.By understanding mathematical series, we grasp how complex functions or datasets can be broken down and analyzed through simpler arithmetical means.

comparison series

A comparison series is a series used to determine the convergence or divergence of another series by comparison. In essence, if you understand the behavior of a known series well, you can use it as a benchmark for other series.In the given exercise, the series \( \sum_{n=1}^{\infty} \frac{1}{n^2} \), which is a known convergent p-series, is used as a comparison series. The idea is:

• The given series \( a_n = \frac{n}{n^3 + 1} \) is compared with \( \frac{1}{n^2} \).
• It is shown that \( \frac{n}{n^3 + 1} < \frac{1}{n^2} \) for sufficiently large \( n \).

If your original series is smaller term-by-term than a known convergent series and all terms are positive, your original series converges as well. This approach is powerful because it leverages previously understood series to infer properties of new, more complex ones.

limit approximation

Limit approximation involves simplifying an expression to more easily understand its behavior as its variable approaches infinity or some other significant point. This is particularly useful in determining series convergence through comparison.In our exercise, notice how:

• The term \( a_n = \frac{n}{n^3 + 1} \) is simplified when \( n^3 + 1 \approx n^3 \).
• This approximation gives \( \frac{n}{n^3} = \frac{1}{n^2} \), linking the problem to a known p-series.

By approximating limits, we can transition from complex terms to simpler, more recognizable forms—allowing us to make informed decisions about convergence and comparison.Understanding limit approximation not only helps in convergence tests but also enhances overall mathematical problem-solving skills by identifying when simplifications are valid and useful.