Question

For each of the following series, use the sequence of partial sums to determine whether the series converges or diverges. $$ \sum_{n=1}^{\infty} \frac{n}{n+2} $$

Short answer

The series diverges.

Step-by-step solution

Understand the series

The series given is \( \sum_{n=1}^{\infty} \frac{n}{n+2} \). This is an infinite series, meaning it has an infinite number of terms to be summed. We are tasked to determine if this series converges (sums to a finite value) or diverges (sums to infinity).

Define the sequence of partial sums

Let \( S_k \) be the k-th partial sum of the series. Thus, \( S_k = \sum_{n=1}^{k} \frac{n}{n+2} \) represents the sum of the first \( k \) terms of the series.

Analyze the terms of the series

The general term of the series \( \frac{n}{n+2} \approx 1 - \frac{2}{n+2} \) as \( n \) becomes very large. This approximation suggests that each term of the series behaves like 1 as \( n \to \infty \).

Check the convergence using the Test for Divergence

According to the Test for Divergence, if \( \lim_{n \to \infty} a_n eq 0 \), the series \( \sum_{n=1}^{\infty} a_n \) diverges. Here, \( a_n = \frac{n}{n+2} \) and \( \lim_{n \to \infty} \frac{n}{n+2} = 1 eq 0 \). This implies that the series diverges.

Key concepts

Sequence of Partial Sums

The sequence of partial sums is a crucial concept for understanding the convergence or divergence of an infinite series. When we deal with a series like \( \sum_{n=1}^{\infty} \frac{n}{n+2} \), the sequence of partial sums \( S_k \) is defined as the sum of the first \( k \) terms. In mathematical terms, this is expressed as: \[ S_k = \sum_{n=1}^{k} \frac{n}{n+2} \] Evaluating the sequence of partial sums involves adding up these terms one by one, and examining the behavior of \( S_k \) as \( k \) becomes very large (approaches infinity). If \( S_k \) approaches a specific finite value, the series converges. If it doesn't settle to any number, the series diverges.

• To determine the behavior, compute several initial sums.
• Check for trends or patterns as \( k \) increases.

Analyzing \( S_k \) helps reveal whether the overall series accumulates to a finite sum or grows indefinitely, giving insight into convergence or divergence.

Convergence Tests

Convergence tests are methods used to determine whether an infinite series converges or diverges. A commonly used test is the Test for Divergence. This test looks at the limit of the general term \( a_n \) of a series as \( n \) approaches infinity. Specifically, the series \( \sum_{n=1}^{\infty} a_n \) diverges if:

• \( \lim_{n \to \infty} a_n eq 0 \).

In our exercise, the given series is \( \sum_{n=1}^{\infty} \frac{n}{n+2} \). Evaluating the limit of \( a_n = \frac{n}{n+2} \) as \( n \to \infty \), we find that: \[ \lim_{n \to \infty} \frac{n}{n+2} = 1 eq 0 \] According to the Test for Divergence, because the terms do not approach zero, the series diverges. Other tests for convergence include the Ratio Test, Root Test, and Integral Test, each with its criteria and applicable scenarios, providing a toolbox of techniques for various series types.

Infinite Series

An infinite series is a sum that extends indefinitely with an infinite number of terms. The series \( \sum_{n=1}^{\infty} \frac{n}{n+2} \) is an example of such a series. Understanding infinite series involves examining the sum of terms beyond just a finite limit. The crucial question with any infinite series is whether it converges to a finite value or not.

• A series converges if the sum approaches a specific number as more terms are added.
• A series diverges if the sum continues to grow or oscillate.

For this series, assessing whether it converges or diverges hinges on understanding the behavior of its terms. The most basic requirement for convergence is that the terms themselves shrink to zero as \( n \) becomes large, which is not the case here. This principle forms the foundation for determining whether an infinite series can sum to a finite value, making it a fundamental concept in analyzing series.