Problem 27 - Single Variable Calculus

Question

Determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty} \frac{n-1}{n^{3}+2}\)

Step-by-Step Solution

Answer

By using the limit comparison test with the convergent p-series \(\sum_{n=1}^{\infty} \frac{1}{n^2}\), we find that the limit of the ratio of the terms is \(L = \frac{1}{1} = 1\). Since L is a finite number and the comparison series is convergent, the given series \(\sum_{n=1}^{\infty} \frac{n-1}{n^{3}+2}\) is also convergent.

Step 1: Identify the given series and the p-series we are comparing it to

The provided series is \(\sum_{n=1}^{\infty} \frac{n-1}{n^3+2}\). We will compare it to the convergent p-series \(\sum_{n=1}^{\infty} \frac{1}{n^2}\) using the limit comparison test.

Step 2: Perform the limit comparison test

To perform the limit comparison test, we will find the limit L of the ratio of their terms as n approaches infinity: \(L = \lim_{n\to\infty} \frac{\frac{n-1}{n^{3}+2}}{\frac{1}{n^2}}\)

Step 3: Simplify the limit

We can simplify this limit by multiplying the numerator by n^2: \(L = \lim_{n\to\infty} \frac{n-1}{n^3+2}\cdot n^2 = \lim_{n\to\infty} \frac{n^3-n^2}{n^3+2}\) Now, we can divide both the numerator and the denominator by \(n^3\): \(L = \lim_{n\to\infty} \frac{1-\frac{1}{n}}{1+\frac{2}{n^3}}\)

Step 4: Evaluate the limit

The limit of this expression is simply the constant value: \(L = \lim_{n\to\infty} \frac{1-\frac{1}{n}}{1+\frac{2}{n^3}} = \frac{1}{1} = 1\) Since L = 1 is a finite number, and the comparison p-series \(\sum_{n=1}^{\infty} \frac{1}{n^2}\) is convergent, the given series \(\sum_{n=1}^{\infty} \frac{n-1}{n^{3}+2}\) is convergent as well.

Step 5: Final Answer

By using the limit comparison test, we conclude that the series \(\sum_{n=1}^{\infty} \frac{n-1}{n^{3}+2}\) is convergent.

Key Concepts

Limit Comparison Test

The Limit Comparison Test is a handy tool in calculus to check the convergence of an infinite series. This method is particularly useful when dealing with complex terms that resemble a simpler series, such as a p-series.

To apply the Limit Comparison Test:

• Identify the series you want to evaluate, called the original series.
• Choose a known series, typically a p-series, for comparison.
• Calculate the limit of the ratio of the terms of your series to the terms of the known series, as the index, often denoted as 'n', goes to infinity.
• If this limit is a positive finite number, both series will either converge or diverge together.

This allows you to draw conclusions about the original series based on the behavior of the known series. In our example, we compared the given series \( \sum_{n=1}^{\infty} \frac{n-1}{n^3+2} \) to the convergent series \( \sum_{n=1}^{\infty} \frac{1}{n^2} \), ending with a limit of 1, indicating convergence.

P-Series

A p-series is a fundamental type of series in calculus, represented generally as \( \sum_{n=1}^{\infty} \frac{1}{n^p} \). The behavior of a p-series is determined by its exponent, 'p'.

Here’s how it works:

• If \( p > 1 \), the series converges, meaning the sum of its terms approaches a finite limit.
• If \( p \leq 1 \), the series diverges, meaning the sum grows indefinitely.

For instance, \( \sum_{n=1}^{\infty} \frac{1}{n^2} \) is a convergent p-series because \( p = 2 \) which is greater than one. This makes p-series a useful benchmark for comparison in the Limit Comparison Test. By comparing other series with known p-series, you can often easily determine convergence or divergence.

Infinite Series

An infinite series is the sum of an endless sequence of numbers. These series are vital in calculus, playing a key role in various mathematical and scientific contexts.

The basic idea is that you keep adding terms, ad infinitum. For example, if you start adding numbers like \( \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots \), you create an infinite series.

To determine if this series converges or diverges, we analyze the behavior of the series as the number of terms grows towards infinity:

• A series is convergent if the partial sums of its terms approach a specific limit.
• A series is divergent if the sums keep growing or oscillating without approaching any fixed value.

Infinite series can represent functions, rates of change, and more, making them a cornerstone of calculus.

Calculus

Calculus is a branch of mathematics focused on change and motion. It comprises two main areas: differential calculus and integral calculus, both of which together form an essential foundation for understanding complex systems.

Key concepts in calculus include:

• Limits: These help us understand behavior as numbers get very large or very small.
• Derivatives: These represent rates of change and are used in differential calculus.
• Integrals: These are used to find areas under curves and accumulate quantities.

In terms of infinite series, calculus helps in determining their convergence through tests such as the Limit Comparison Test. Calculus extends beyond simple computations to modeling and analyzing real-world phenomena, making it crucial for sciences, engineering, economics, and more.