Problem 26 - Essential Calculus

Question

Use Theorem 7.11 to determine the convergence or divergence of the \(p\) -series. $$ \sum_{n=1}^{\infty} \frac{1}{n^{\pi}} $$

Step-by-Step Solution

Answer

The series \( \sum_{n=1}^{\infty} \frac{1}{n^{\pi}} \) is a convergent p-series since its power \( \pi > 1 \).

Step 1: Identify the p-Series

The given series is a p-series, because it has the form \( \sum_{n=1}^{\infty} \frac{1}{n^p} \) where \( n \) represents the term number and \( p \) is a constant. In this case, \( p = \pi \). Now it's important to notice that the value of \(\pi\) is approximately 3.14.

Step 2: Apply the p-Series Convergence Test

Using Theorem 7.11, this p-series will converge if \( p > 1 \) and diverge if \( p ≤ 1 \). Here, \( p = \pi \) is approximately 3.14, which is greater than 1.

Step 3: Determine convergence or divergence

Since \( \pi > 1 \), by Theorem 7.11, it can be inferred that the series \( \sum_{n=1}^{\infty} \frac{1}{n^{\pi}} \) converges.

Key Concepts

Theorem 7.11

In the study of series in mathematics, Theorem 7.11 plays a crucial role, especially when dealing with p-series. This theorem provides a simple yet powerful criterion to determine whether a given p-series converges or diverges. Specifically, it states that a p-series of the form \[\sum_{n=1}^{\infty} \frac{1}{n^p} \]will converge if the exponent \( p \) is greater than 1, and diverge if \( p \) is less than or equal to 1. This clear-cut guideline is immensely useful because it allows us to decide the behavior of the series just by observing the value of \( p \).

It's important to understand that this theorem is applicable to p-series, characterized by the form \( \frac{1}{n^p} \) where \( n \) is an integer sequence starting from 1, and \( p \) is a constant. This consistency in form makes Theorem 7.11 a straightforward yet powerful tool in series analysis.

series convergence

Series convergence is a fundamental concept in mathematical analysis. It refers to whether a series settles into a specific value as we sum its terms across an infinite span. For a series \( \sum a_n \), if the partial sums \( S_N = a_1 + a_2 + \ldots + a_N \) approach a finite limit as \( N \) increases, the series is said to converge.

• This means that as you add more terms, you get closer to some fixed number.
• If there's no such finite number, the series diverges.

In the context of p-series, convergence largely depends on the exponent \( p \). For the given series \( \sum_{n=1}^{\infty} \frac{1}{n^{\pi}} \), where \( \pi \approx 3.14 \), the series converges according to Theorem 7.11 because \( \pi > 1 \). This exemplifies how knowing the value of \( p \) helps us understand the series convergence quickly.

mathematical divergence

When we talk about mathematical divergence, it's essentially the opposite of convergence. Here, a series fails to approach a particular finite limit. The series might grow indefinitely, or oscillate without settling at any fixed value. This behavior can be identified based on certain criteria, such as those given by tests like Theorem 7.11.

For instance, if you have a p-series \( \sum_{n=1}^{\infty} \frac{1}{n^p} \) with \( p \leq 1 \), the terms do not reduce fast enough to sum up to a finite value. As a result, the series diverges.

• A classic example is the harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \), where \( p = 1 \), which is well-known to diverge.
• This contrasts with cases where \( p > 1 \), leading to convergence as previously discussed.

Understanding the conditions that cause divergence helps us predict and analyze the behavior of different types of series, and it is a topic of great importance in mathematical studies.