Problem 11 - Calculus: One and Several Variables

Question

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

Step-by-Step Solution

Answer

The given series is divergent.

Step 1: Identifying the structure

First, recognize that the series is a p-series, where the general form of the series is \(\sum\)\(1/n^p\), where 'n' is the number of terms and 'p' is a real number. Here, p is 2/3.

Step 2: Apply the p-series test

To determine whether the p-series is convergent or divergent, apply the p-series test, which states: For a series to converge, 'p' must be greater than 1. If 'p' is less than or equal to 1, then the series diverges.

Step 3: Compare p

Compare the value of p, in this case, which is 2/3, with 1. As 2/3 is less than 1, indicating the series should be divergent.

Key Concepts

Series Convergence

When discussing series convergence, our main goal is to determine whether the sum of an infinite series approaches a finite number. A series is a sum of terms of a sequence, and convergence means that as we add more terms, the series settles down to a specific value.

Convergence is important because it tells us about the behavior of a series in the long run.

• A convergent series has a finite limit.
• As you add more terms, the total gets closer to a particular number.

When a series converges, it is extremely valuable in mathematics because it allows complex infinite processes to be understood as finite limits. This has applications in various fields like physics, engineering, and computer science. To check if a series converges, mathematicians use various tests, such as the p-series test, which is particularly useful for series with terms including exponents.

Divergent Series

The opposite of convergence is divergence. A divergent series is one that does not approach a specific number as we add more terms. Instead, it keeps growing or oscillating without stability.

Divergent series have their own challenges and opportunities.

• An infinite sum that does not settle to a limit is divergent.
• In some cases, divergent series can be manipulated to provide useful information under specific conditions.

In practice, determining whether a series diverges helps in identifying limits to calculations or approximations. It's crucial, as it avoids relying on sums that will not yield stable or meaningful results. For the p-series \sum \frac{1}{k^{2/3}}, it diverges due to its exponent being \(<1\), signaling the sum does not have a finite limit.

P-Series

The p-series is a fundamental concept in sequences and series. It is a series of the form \(\sum \frac{1}{n^p}\), where \(p\) is a constant and \(n\) is a positive integer.

Understanding p-series is vital because of their predictability and straightforward nature.

• If \(p > 1\), the series converges, meaning it reaches a particular value as \(n\) increases.
• If \(p \leq 1\), the series diverges, indicating that the series does not settle down to a finite sum.

This test highlights critical behavior in series and helps simplify complex mathematical problems. It's essential in verifying the convergence or divergence when dealing with series that look similar but have different exponents. In the given exercise, applying the p-series test to the series \sum \frac{1}{k^{2/3}}, it was determined to diverge as the value of \(p\) is \(2/3\).