Problem 15 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Do the following. i. Write out the rst two terms of the series. ii. Determine whether or not the series converges. iii. If the series converges, determine its sum. $$ \sum_{n=100}^{\infty} \frac{10}{n} $$

Step-by-Step Solution

Answer

The first two terms of the series are 0.1 and approximately 0.099. The series diverges and thus, does not have a sum.

Step 1: Write the first two terms

Start by finding the first two terms of the series. Substitute \(n=100\) and \(n=101\) into the series function: \[First Term = \frac{10}{100} = 0.1\] \[Second Term = \frac{10}{101} ≈ 0.099\] So the first two terms are 0.1 and approximately 0.099.

Step 2: Determine convergence or divergence

The series is a P-Series, which has the form \(\sum_{n=1}^{\infty} \frac{1}{n^p}\), and it converges if \(p > 1\), and diverges if \(p ≤ 1\). In this case, it is a 1-Series \(\sum_{n=100}^{\infty} \frac{10}{n}\). Our \(p\) is equal to 1, which means according to the P-Series test, our series diverges.

Step 3: Find the sum

Since the series was proven to diverge, it does not have a sum to find.

Key Concepts

Convergence and Divergence

Convergence and divergence are essential when analyzing series. In mathematics, a series is the sum of the elements of a sequence. Often, we want to know if this sum approaches a specific value as more and more terms are added. If it does, we say the series converges. If it does not approach any particular value, even as we add infinitely many terms, the series diverges.
To determine if a series converges or diverges, we must examine its terms and how they behave as the index goes to infinity. For a series given by \( \sum_{n=a}^{\infty} a_n \), we look at the sum of terms starting from index \(a\) approaching infinity.

• If the series approaches a finite number, it converges.
• If the series increases indefinitely or does not settle on a number, it diverges.

Understanding convergence is crucial to determining further properties of the series, like finding its sum when it converges.

P-Series Test

The P-Series test is a straightforward but powerful tool for determining the convergence of a series. A P-Series is defined as \( \sum_{n=1}^{\infty} \frac{1}{n^p} \), where \( p \) is a positive constant. The convergence of a P-Series depends directly on the value of \( p \):

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

Let's break this down further. As the exponent \( p \) determines how fast the terms \( \frac{1}{n^p} \) decrease, larger values of \( p \) lead to rapidly decreasing terms, thus a finite sum. When \( p = 1 \), the terms decrease more slowly, resembling the harmonic series which famously diverges. The problem earlier uses a P-Series with \( p = 1 \) (since the term \( \frac{10}{n} \) is equivalent to \( \frac{10}{1} \times \frac{1}{n^1} \)). Thus, it diverges, confirming that the sum does not converge to a specific value.

Sum of Series

The sum of a series is a fundamental concept when working with sequences. When a series converges, it approaches a specific value, known as its sum. However, it's essential to remember that not every series has a sum. This only occurs when the series converges. In the context of our earlier problem, since the series diverges due to \( p = 1 \), finding the sum is not applicable.
Why is the sum important? It often represents quantities in the real world, like total time, distance, or accumulated interest. A convergent series can give us a valuable finite number, aiding calculations in fields like science and engineering. When we determine if a series converges and find its sum, we're not only solving abstract problems but also preparing for these real-world applications. Notably, techniques for finding sums may involve comparisons, transformations, or even advanced calculus tools for more complicated series.