Problem 35 - Thomas Calculus

Question

Which of the sequences \(\left\\{a_{n}\right\\}\) in Exercises \(27-90\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=1+(-1)^{n} $$

Step-by-Step Solution

Answer

The sequence diverges; it oscillates between 0 and 2.

Step 1: Identify the Pattern in the Sequence

The sequence given is \( a_{n} = 1 + (-1)^{n} \). We need to identify the behavior of this sequence as \( n \) increases. Notice that for even values of \( n \) (e.g., \( n = 2, 4, 6, \ldots \)), \((-1)^{n} = 1\). For odd values of \( n \) (e.g., \( n = 1, 3, 5, \ldots \)), \((-1)^{n} = -1\). This means the sequence alternates between two values.

Step 2: Determine Specific Values of the Sequence

Let's determine the first few terms of the sequence to see the pattern: For \( n=1 \), \( a_{1} = 1 + (-1)^1 = 0 \). For \( n=2 \), \( a_{2} = 1 + (-1)^2 = 2 \). For \( n=3 \), \( a_{3} = 1 + (-1)^3 = 0 \), and so on. The sequence alternates between 0 and 2: \( 0, 2, 0, 2, \ldots \).

Step 3: Analyze the Convergence of the Sequence

For a sequence to be convergent, it must approach a specific finite limit as \( n \) approaches infinity. In this case, the terms of the sequence do not settle towards a single value but instead continually oscillate between 0 and 2. This indicates that \( a_{n} \) does not converge to a limit.

Step 4: Conclusion on Convergence or Divergence

As the sequence \( a_{n} = 1 + (-1)^n \) does not approach a particular single value, it is divergent. This alternation between 0 and 2 means the limits from the terms do not settle to a single number.

Key Concepts

Alternating Sequences

Alternating sequences, like the one given by \( a_n = 1 + (-1)^n \), are sequences that switch back and forth between two or more values as \( n \) increases. In this exercise, the sequence alternates between the values 0 and 2 for different values of \( n \).

When you look at it closely, for even numbers of \( n \), the term \( (-1)^n \) becomes 1, resulting in \( a_n = 2 \).
For odd numbers of \( n \), \( (-1)^n \) turns into -1, making \( a_n = 0 \).

This kind of alternation is a characteristic feature of alternating sequences.

• Even \( n \): The sequence value is 2.
• Odd \( n \): The sequence value is 0.

Understanding this behavior is crucial when trying to determine the convergence properties of a sequence.

Limit of a Sequence

The limit of a sequence is the value that the sequence approaches as the number of terms goes to infinity.
In other words, it's the value that the terms of the sequence get closer and closer to as \( n \) gets larger and larger.
For a sequence to have a limit, it must settle down to one particular number or value as \( n \) increases.

In our specific sequence \( a_n = 1 + (-1)^n \), the terms do not get closer to any single value.
Instead, the sequence fluctuates between the values 0 and 2.
This means that the sequence does not have a limit because:

• There is no single number that all terms of the sequence approach.
• The terms of the sequence continue to oscillate indefinitely.

Therefore, understanding the limit of a sequence is key in determining whether a given sequence converges.

Divergent Sequences

A divergent sequence is one that does not settle to a single value as \( n \) approaches infinity.
In the case of the sequence \( a_n = 1 + (-1)^n \), it is an example of a divergent sequence because it alternates endlessly between two values (0 and 2).

Divergent sequences can exhibit many different patterns, but a common sign of divergence is the non-existence of a limit.
Here are some characteristics of divergent sequences like this one:

• The sequence does not approach a fixed value.
• The oscillation between two or more values continues indefinitely.
• As \( n \) increases, the sequence does not stabilize.

Recognizing divergent sequences is essential in analytical assessments, as they behave differently from convergent sequences and often require distinct approaches in mathematical analysis.