Problem 26 - Precalculus: Mathematical for Calculus

Question

If the sequence is convergent, find its limit. If it is divergent, explain why. $$a_{n}=\cos n \pi$$

Step-by-Step Solution

Answer

The sequence is divergent because it alternates between 1 and -1 without approaching a single value.

Step 1: Understand the Sequence

The given sequence is \( a_n = \cos(n \pi) \). We need to analyze the behavior of this sequence to determine if it converges or diverges.

Step 2: Plug in Initial Values

Calculate the first few terms to identify any pattern:- For \( n = 0 \), \( a_0 = \cos(0 \cdot \pi) = \cos(0) = 1 \).- For \( n = 1 \), \( a_1 = \cos(\pi) = -1 \).- For \( n = 2 \), \( a_2 = \cos(2\pi) = 1 \).- For \( n = 3 \), \( a_3 = \cos(3\pi) = -1 \).- For \( n = 4 \), \( a_4 = \cos(4\pi) = 1 \).The pattern alternates between 1 and -1.

Step 3: Identify the Pattern

Notice that for even \( n \), \( a_n = 1 \), and for odd \( n \), \( a_n = -1 \). This means the sequence alternates between 1 and -1 as \( n \) increases.

Step 4: Determine Convergence or Divergence

For a sequence to be convergent, its terms must approach a single number as \( n \) approaches infinity. In this case, because the sequence \( a_n \) alternates between 1 and -1 and does not approach a single value, the sequence is divergent.

Key Concepts

Sequence Convergence

In mathematics, a sequence is said to be convergent if the terms of the sequence come closer and closer to a specific value as the sequence progresses. This specific value is called the 'limit' of the sequence. If a sequence is convergent, this limit exists and can be calculated. To better understand the concept of convergence, imagine a sequence as a set of instructions taking you closer to a specific place. If each step takes you nearer to this place, then your steps are converging. Similarly, if the numbers in a sequence steadily move towards a particular limit, that sequence is deemed convergent. Divergent sequences, on the other hand, do not settle towards any single value. In such sequences, the numbers might increase indefinitely, oscillate between numbers, or behave chaotically. Like with the sequence given in the exercise, each number keeps swinging between 1 and -1 without ever nearing a single stopping point. This signifies divergence, as the sequence lacks a unique destination value, implying that it does not converge.

Limit of a Sequence

The limit of a sequence refers to the value that the terms of a convergent sequence tend to as the number of terms goes to infinity. When a sequence has a limit, this means that the terms eventually get as close as desired to this particular value and remain close as more terms are added.Mathematically, we define the limit of a sequence, say \( a_n \), as \( L \) if for every positive number \( \epsilon \), there exists an integer \( N \) such that for all integers \( n > N \), the absolute difference between \( a_n \) and \( L \) is smaller than \( \epsilon \):

• \( |a_n - L| < \epsilon \) for \( n > N \).

This definition helps convey the idea that after a certain point, every term of the sequence is within an arbitrarily small distance from \( L \).For sequences like \( a_n = \cos(n \pi) \), however, each term oscillates between values instead of approaching a limit, making it impossible to establish a single number that all terms tend towards. Therefore, this sequence does not have a limit.

Behavior of Sequences

Understanding the behavior of sequences is crucial in identifying whether they converge or diverge. The behavior is essentially how the terms change as the sequence progresses towards infinity.Sequences can:

• Converge: Terms get closer to a single value.
• Diverge: Terms fail to settle at any single value.
• Oscillate: Terms alternate between certain values, as in our exercise, moving between 1 and -1.
• Grow Unbounded: Terms increase or decrease without bound.

Analyzing the behavior allows us to classify the sequence. Like in this exercise, the sequence oscillates endlessly between 1 and -1. Its alternating pattern, based on whether \( n \) is even or odd, depicts a divergent behavior, as there's no single value being approached.By observing the changes in sequence terms, we can develop a deeper understanding of whether a sequence will eventually stabilize at a specific value or continue acting unpredictably, as seen when terms change erratically without a clear trend.