Problem 26 - Thomas Calculus

Question

Which of the series converge, and which diverge? Use any method, and give reasons for your answers. \begin{equation}\sum_{n=1}^{\infty} \frac{1}{\sqrt{n^{3}+2}}\end{equation}

Step-by-Step Solution

Answer

The series converges by the Comparison Test with the convergent p-series \( \sum \frac{1}{n^{3/2}} \).

Step 1: Determine Test Choice

To determine if the series \( \sum_{n=1}^{\infty} \frac{1}{\sqrt{n^{3}+2}} \) converges or diverges, we need to choose an appropriate convergence test. Since the series involves a radical function of \( n \) (namely \( \sqrt{n^3 + 2} \)), we shall use the Comparison Test by comparing it to a simpler series.

Step 2: Compare with a Simpler Series

We note that for large \( n \), the expression \( \sqrt{n^3 + 2} \) can be approximated by \( \sqrt{n^3} = n^{3/2} \). Thus, we compare with the series \( \sum_{n=1}^{\infty} \frac{1}{n^{3/2}} \), which is a known p-series.

Step 3: Test Known Series for Convergence

The series \( \sum_{n=1}^{\infty} \frac{1}{n^{3/2}} \) is a p-series where \( p = 3/2 > 1 \). It is a known fact that p-series converge if \( p > 1 \). Thus, \( \sum_{n=1}^{\infty} \frac{1}{n^{3/2}} \) converges.

Step 4: Apply the Comparison Test

Since \( \sqrt{n^3 + 2} > n^{3/2} \) for all \( n \geq 1 \), it follows that \( \frac{1}{\sqrt{n^3 + 2}} < \frac{1}{n^{3/2}} \). By the Comparison Test, because the larger series \( \sum_{n=1}^{\infty} \frac{1}{n^{3/2}} \) converges, our original series \( \sum_{n=1}^{\infty} \frac{1}{\sqrt{n^{3}+2}} \) also converges.

Key Concepts

Understanding the Comparison Test

The comparison test is an analytical method used to determine the convergence or divergence of infinite series. It involves comparing the series in question with another series whose convergence properties are already known.

The idea is simple: if you can show that your series behaves similarly to a well-understood series, you can draw conclusions about its convergence or divergence.

• If series A is always less than or equal to series B, and series B converges, then series A also converges.
• Conversely, if series A is greater than series B, and B diverges, then A diverges as well.

In our example, the given series involves terms of the form \(\frac{1}{\sqrt{n^3 + 2}}\). By approximating \(\sqrt{n^3 + 2}\) as \(n^{3/2}\), we compare to the known convergent p-series \(\sum \frac{1}{n^{3/2}}\). Since \(\frac{1}{\sqrt{n^3+2}} < \frac{1}{n^{3/2}}\), the original series converges by the comparison test.

Exploring p-series

P-series are a specific type of series given by the formula \(\sum_{n=1}^{\infty} \frac{1}{n^p}\), where \(p\) is a real number. Understanding p-series is crucial when studying the convergence of sequences and series.

The convergence of p-series can be simply determined:

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

This rule is essential for analyzing other series by comparison. In the solution above, the comparison was made with a p-series where \(p = 3/2\). Since \(3/2 > 1\), we concluded that the comparison series converges, helping us determine the behavior of our original series.

Understanding Infinite Series

An infinite series is the sum of an infinite sequence of numbers. The study of these series is vital in mathematics as they appear in calculus, number theory, and many areas of science.

Not all infinite series sum to a finite number, which is why determining convergence or divergence is important. When a series converges, its terms approach a limiting value, allowing the series to sum to a finite number.

• Convergent series are stable and useful in many practical applications.
• Divergent series grow without bound and typically indicate instability or lack of a solution in the context applied.

Analyzing infinite series often involves various tests, such as the comparison test, to assess these behaviors.

Sequence and Series Convergence

The convergence of a sequence or series indicates that it approaches a specific value as \(n\) (the number of terms) approaches infinity.

Sequences are collections of numbers, while series are sums of sequences. The convergence test for a series checks if adding more terms leads us to a stable sum.

• A converging sequence will have its terms getting closer and closer to a particular number.
• For series, convergence implies that the sum of its terms reaches a limit, even with infinitely many terms.

Understanding the criteria for convergence, like the comparison test or observing behavior akin to convergent p-series, is essential for navigating more complex mathematical problems.