Problem 11 - Essential Calculus Early Transcendentals

Question

Determine whether the series is convergent or divergent. $$\sum_{n=1}^{\infty} \frac{2}{n^{0.85}}$$

Step-by-Step Solution

Answer

The series is divergent.

Step 1: Identify the Type of Series

The given series is of the form \( \sum_{n=1}^{\infty} a_n \), where \( a_n = \frac{2}{n^{0.85}} \). This resembles a p-series \( \sum_{n=1}^{\infty} \frac{1}{n^p} \). The general p-series converges if \( p > 1 \) and diverges if \( p \leq 1 \).

Step 2: Compare with the p-Series Test

Compare the given series \( \sum_{n=1}^{\infty} \frac{2}{n^{0.85}} \) with a p-series. Here, the exponent attached to \( n \) in the denominator is \( 0.85 \), so the series is similar to \( \sum_{n=1}^{\infty} \frac{1}{n^{0.85}} \).

Step 3: Determine Convergence or Divergence

Apply the p-series test: The exponent \( 0.85 \) is less than 1. According to the p-series test, if \( p \leq 1 \), the series diverges. Since \( 0.85 < 1 \), the series \( \sum_{n=1}^{\infty} \frac{2}{n^{0.85}} \) diverges.

Key Concepts

p-series test

When we deal with infinite series in mathematics, the **p-series test** is a vital tool to help us decide if a series converges or diverges. A p-series is typically written as \( \sum_{n=1}^{\infty} \frac{1}{n^p} \). Here, the core idea is to observe the value of the exponent \( p \). This value determines whether the series behaves itself, that is, converges, or misbehaves, leading to divergence.

According to the p-series test:

• The series converges if \( p > 1 \).
• The series diverges if \( p \leq 1 \).

Let's think of it this way: the larger the value of \( p \), the faster the terms get smaller, making it easier for the series to converge. When \( p \leq 1 \), terms don’t shrink rapidly enough, leading the series to run off to infinity, hence a divergence. The p-series test is a quick and efficient way to deal with such problems.

divergent series

A **divergent series** is an important concept in calculus which occurs when the sum of a series grows towards infinity rather than settling at a fixed value. This means the more terms you add up in the series, the further away it gets from a specific sum. Divergence often indicates that the series doesn’t converge to a limit, meaning you can’t pin down one finite number it approaches.

In our example \( \sum_{n=1}^{\infty} \frac{2}{n^{0.85}} \), since the exponent \( 0.85 \) is less than 1, the series diverges. Intuitively, the terms \( \frac{2}{n^{0.85}} \) don't decrease fast enough as \( n \) grows. They might get smaller, but not rapid enough for their cumulative sum to cap off at a finite number.

Understanding divergent series is crucial, as it often helps in distinguishing between calculable and incalculable phenomena in both mathematics and real-world applications, ensuring we don't make unrealistic assumptions when working with mathematical models.

convergence tests

The term **convergence tests** refers to various techniques used in mathematics to check if an infinite series converges or diverges. Having different tests allows you to choose the suitable method depending on the series you are handling. Let's look at a few commonly used convergence tests:

• p-series test: As discussed, it works best for series of the type \( \sum_{n=1}^{\infty} \frac{1}{n^p} \).
• Ratio test: Good for complex series involving factorials or exponential terms.
• Root test: Another potent tool, useful when dealing with roots of term expressions.

When using convergence tests, it’s key to match the right test to your series type. Each test provides a framework to decipher the behavior of the series. The ability to differentiate between convergence and divergence ensures proper analysis and significant conclusions in mathematical computations. Remember, a correct identification via a convergence test makes all the difference in understanding the subtleties of infinite series.