Problem 27 - Calculus: One and Several Variables

Question

Determine whether the series converges or diverse. $$\sum \frac{2 k+1}{\sqrt{k^{3}+1}}$$

Step-by-Step Solution

Answer

The series diverges.

Step 1: Identify Test

Because the series possesses both positive terms and an apparent end behavior, the Comparison Test seems applicable.

Step 2: Identify Comparable series

Next is to find a series to compare. In this case, if you simplify the terms of the series by removing lower order terms, you get \( a_n = \frac{2n}{n^{3/2}} = \frac{2}{n^{1/2}} \). So it is cleared that the series \( \sum \frac{2}{n^{1/2}} \) can be a good series to compare the given series to.

Step 3: Apply Comparison Test Criteria

For all \( k \geq 1 \) in the function, \( \frac{2k+1}{\sqrt{k^{3}+1}} \leq \frac{2k}{k^{3/2}} = \frac{2}{k^{1/2}} \) which means that for all \( k \geq 1 \), the terms of the given series are less than or equal to the terms in the series \( \sum \frac{2}{k^{1/2}} \) which is a p-series with \( p=1/2 \). Considering that p-series will only converge if \( p > 1 \) and here \( p=1/2<1 \). So, the comparison series diverges.

Step 4: Conclude if Series Converges or Diverges

Since the comparison series diverges according to the Comparison Test, the original series \( \sum \frac{2k+1}{\sqrt{k^{3}+1}} \) also diverges.

Key Concepts

Comparison Test

The Comparison Test is a useful tool for determining the convergence or divergence of a series. When you have a series with positive terms, you can compare it to a second, perhaps simpler, series that you already understand well. When using the Comparison Test:

• If the series you know (the comparison series) converges, and every term in your original series is smaller than the corresponding term in the comparison series, your original series converges too.
• Conversely, if your comparison series diverges and every term in your original series is larger than the corresponding term in the comparison series, then your original series diverges.

This test is particularly useful when there is an obvious way to simplify your original series into a familiar form. In the case of this exercise, a comparison was made to a simpler p-series. Understanding how to apply the Comparison Test can save much time in verifying convergence or divergence of complex series.

p-series

A p-series is a specific type of series that looks like \[ \sum \frac{1}{n^p} \]where \( p \) is a constant. The p-series is an important and easily recognizable series within convergence tests. It behaves in a simple manner:

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

In the original exercise, our transformed series \( \sum \frac{2}{k^{1/2}} \) is effectively a p-series where \( p = 1/2 \). Since \( p = 1/2 < 1 \), the series diverges according to the rules above. Recognizing a p-series helps quickly ascertain the behavior of a series by matching it to known convergence patterns.

Convergence and Divergence Tests

Convergence and divergence tests form the backbone of understanding series' behavior, determining whether series approach a finite sum or not. Apart from the Comparison Test and recognition of p-series, other tests are available:

• Ratio Test: Utilizes the ratio of consecutive terms to establish convergence.
• Root Test: Examines the nth root of terms for similar purposes.
• Integral Test: Compares a series to an integral to deduce convergence.

These test provide different tools for different series types. Often, a single test offers the simplest path to the answer, but sometimes multiple tests can interrelate insights about a complex series. The goal of these tests is to provide clear criteria that can be applied to the series structure itself, offering efficient identification of convergence or divergence without the arduous calculation of partial sums.