Question

In Exercises 33-52, determine the convergence or divergence of the sequence with the given \(n\) th term. If the sequence converges, find its limit. \(a_{n}=(-1)^{n}\left(\frac{n}{n+1}\right)\)

Short answer

The sequence \(a_{n}=(-1)^{n}\left(\frac{n}{n+1}\right)\) is divergent and does not have a definite limit.

Step-by-step solution

Identify the Sequence

The sequence that is given is \(a_{n}=(-1)^{n}\left(\frac{n}{n+1}\right)\).

Examine for Divergence

A sequence is said to be divergent if as \(n \rightarrow \infty\), the sequence does not tend to a limit. If we substitute \(n \rightarrow \infty\) into \(a_{n}\), we obtain \((-1)^{\infty}\left(\frac{\infty}{\infty+1}\right)\). Here, \((-1)^{\infty}\) does not have a definite limit because the sign of the term keeps changing. Since neither positive nor negative infinity is an acceptable limit, we call this sequence divergent.

Identify the Limit

A sequence is said to have a limit if as \(n \rightarrow \infty\), the terms of the sequence approach a definite number. As defined above, the sequence does not have a definite limit as it flips signs for every new term.

Key concepts

Alternating Sequence

An alternating sequence is characterized by its terms switching signs between positive and negative with each subsequent term. In the exercise provided, the sequence is defined as \( a_n = (-1)^n\left(\frac{n}{n+1}\right) \), which is a classic example of an alternating sequence due to the presence of the \( (-1)^n \) term.

When looking at alternating sequences, one of the key features to consider is whether or not the sequence converges. A useful test for convergence with alternating sequences is the Alternating Series Test. However, in this case, the key aspect is that the \( (-1)^n \) factor does not converge as \( n \) approaches infinity, which leads to the conclusion that this sequence does not have a limit and, therefore, is divergent.

Limits of Sequences

The limit of a sequence is a fundamental concept in calculus and analysis, which deals with what value the terms of a sequence approach as the index \( n \) goes to infinity. If a sequence has a limit, then it is considered to be convergent; otherwise, it is divergent. The limit, if it exists, must be a single, fixed number.

To find the limit of a sequence, we examine the behavior of its nth term as \( n \) increases indefinitely. In the given exercise with the sequence \( a_n = (-1)^n \left(\frac{n}{n+1}\right) \), the fraction \( \frac{n}{n+1} \) approaches 1 as \( n \) approaches infinity. However, due to the alternating factor \( (-1)^n \), the limit of the entire sequence is not a fixed number but rather fluctuates, which means the sequence does not converge.

Divergent Sequence

A divergent sequence is one where the terms do not converge to a fixed limit as the index increases. Divergence occurs when the terms of a sequence display unbounded behavior, or when the terms oscillate between values without settling down to a single value. In the context of the given exercise, the sequence \( a_n = (-1)^n \left(\frac{n}{n+1}\right) \) is divergent.

This can be intuitively understood because the term \( (-1)^n \) causes each term of the sequence to alternate in sign, resulting in no definite limit. Additionally, as explained in Step 2 of the solution, substituting \( n \rightarrow \infty \) into the sequence does not yield a fixed number but an expression that fluctuates, which is characteristic of a divergent sequence.