Problem 39 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

The sequence \( a_n = 1 + (-1)^n \) diverges.

Step 1: Understand the Sequence

Let's look at the given sequence, \( a_{n} = 1 + (-1)^{n} \). This sequence alternates between values depending if \( n \) is even or odd.

Step 2: Sequence Pattern Analysis

Evaluate the pattern by considering the parity of \( n \).- If \( n \) is even (e.g., \( n = 2 \)), \((-1)^{n} = 1\), so \( a_{n} = 1 + 1 = 2 \).- If \( n \) is odd (e.g., \( n = 1 \)), \((-1)^{n} = -1\), so \( a_{n} = 1 - 1 = 0 \).Thus, the sequence alternates between \( 2 \) and \( 0 \).

Step 3: Determine Convergence or Divergence

A sequence converges if it approaches a single specific value as \( n \) tends to infinity. Since our sequence is alternating between two different values (\( 0 \) and \( 2 \)), it does not settle to a single value as \( n \) increases. Therefore, the sequence diverges.

Key Concepts

Sequence Divergence

In mathematics, a sequence is said to diverge if it does not approach a single fixed value as its terms progress to infinity. For the sequence given by \( a_{n} = 1 + (-1)^{n} \), understanding divergence is key because this sequence does not converge to a particular number. Instead, it keeps oscillating between two values: 0 and 2.

• When \( n \) is odd, \[ a_{n} = 0 \].
• When \( n \) is even, \[ a_{n} = 2 \].

As you can see, as \( n \) increases, the terms don't narrow down towards a single number, indicating that this sequence is divergent. A clear way to determine this is by observing that these alternating values do not approach any single finite limit.Since it cannot settle into convergence (a single steady value), we classify this behavior as divergence.

Alternating Sequences

Alternating sequences are sequences in which the signs or patterns of numbers alternate in some regular fashion. In our exercise, the sequence \( a_{n} = 1 + (-1)^{n} \) is an excellent example because it alternates based on the parity of \( n \).When \( n \) is even, the sequence produces one result (2), and when \( n \) is odd, it yields another (0). Such sequences are noteworthy because they do not move towards convergence automatically and need careful examination.

• Every other term differs due to the alternating factor \((-1)^{n}\).
• This can lead to oscillations as seen here between 0 and 2.

Understanding alternating sequences helps grasp why certain patterns do not converge and how their structure, based on alternating terms, can inherently lead to divergence.

Limit of a Sequence

The limit of a sequence is essentially the value that the terms of the sequence approach as the index \( n \) becomes very large. In other words, if the terms of the sequence steadily get closer to some fixed numerical value, we say the sequence converges to that limit.However, for our sequence \( a_{n} = 1 + (-1)^{n} \), no such limit exists because the sequence does not consistently approach a single value. Instead, its alternating nature prevents it from getting sufficiently close to any number.

• If a sequence has a limit, it is a convergent sequence.
• For a diverging sequence like ours, discussing a limit isn’t applicable since it lacks a single point of convergence.

This distinction is crucial to understanding sequences as it marks the difference between convergent behavior (approaching a limit) versus divergent behavior (failing to approach any particular value). Knowing whether a limit exists helps characterize the sequence's nature and future behavior.