Problem 73 - Essential Calculus

Question

Let \(a_{n}=\frac{n+1}{n}\). Discuss the convergence of \(\left\\{a_{n}\right\\}\) and \(\sum_{n=1}^{\infty} a_{n}\).

Step-by-Step Solution

Answer

The limit of the sequence \(a_{n}\) as \(n\) approaches infinity is 1, and the series \(\sum_{n=1}^{\infty} a_{n}\) diverges.

Step 1: Determine the Sequence Limit

In the first part, to find the limit of the sequence \(a_{n}\) as \(n\) approaches infinity, divide each term in the fraction by \(n\). The sequence becomes: \(a_{n}=1+\frac{1}{n}\). As \(n\) approaches infinity, \(\frac{1}{n}\) approaches 0. Therefore, the limit of the sequence \(a_{n}\) as \(n\) approaches infinity is 1.

Step 2: Check the Infinity Series Convergence

To check the convergence of the series \(\sum_{n=1}^{\infty} a_{n}\), apply the nth Term Test. The nth Term Test states that if \(\lim_{{n \to \infty}} a_{n} ≠ 0\), then the series \(\sum_{n=1}^{\infty} a_{n}\) diverges. Since, from Step 1, \(\lim_{{n \to \infty}} a_{n} = 1 ≠ 0\), this implies that the series \(\sum_{n=1}^{\infty} a_{n}\) diverges.

Key Concepts

Convergence of Sequences

The convergence of a sequence is a fundamental concept in calculus and involves understanding the behavior of the sequence's terms as the index approaches infinity. A sequence \( \{a_n\} \) converges if it approaches a specific, finite number as \( n \) becomes very large. This number is called the limit of the sequence. For instance, consider the sequence \( a_n = \frac{n+1}{n} \). To examine its convergence, we simplify it to \( a_n = 1 + \frac{1}{n} \). As \( n \) grows, the term \( \frac{1}{n} \) gets closer and closer to zero, implying the entire sequence \( a_n \) converges to \( 1 \). To understand better, remember these key ideas:

• A sequence has a limit \( L \) if, for every positive number \( \epsilon \), however small, there's a point in the sequence where all subsequent terms are within \( \epsilon \) of \( L \).
• A sequence that approaches a specific value is a converging sequence.
• If a sequence doesn't converge to any value, it's divergent.

Understanding these helps predict patterns in sequences.

Nth Term Test

The nth term test is a handy tool used to determine if a series can potentially converge or must diverge based solely on the terms of the sequence. It's a straightforward initial step in analyzing series. Generally, for a series \( \sum_{n=1}^{\infty} a_n \), the nth term test states:

• If \( \lim_{{n \to \infty}} a_n eq 0 \), then the series \( \sum_{n=1}^{\infty} a_n \) is guaranteed to diverge.
• If \( \lim_{{n \to \infty}} a_n = 0 \), the series might converge, but it doesn't automatically mean it does. Further tests are needed.

This test is particularly useful for quickly ruling out the possibility of convergence if the limit doesn't tend toward zero. For the sequence \( a_n = \frac{n+1}{n} \), its limit, as found earlier, is 1. Since 1 is not zero, the series \( \sum_{n=1}^{\infty} a_n \) diverges by the nth term test.

Divergence of Series

Understanding the divergence of a series is essential to analyzing infinite sums. When a series diverges, it implies that the sum does not approach a particular finite value as the series progresses infinitely. For a series \( \sum_{n=1}^{\infty} a_n \) to be divergent, the sequence of its partial sums must not converge to any finite number. Key points to remember include:

• If the terms of a series do not approach zero, the series cannot converge.
• In many cases, even if the terms do approach zero, the series can still diverge due to the behavior of the partial sums.

Applying this to our example, since \( \lim_{{n \to \infty}} a_n = 1 eq 0 \), the series \( \sum_{n=1}^{\infty} \frac{n+1}{n} \) diverges. It's crucial to recognize these patterns to effectively determine the nature of series in calculus.