Problem 20 - Essential Calculus Early Transcendentals

Question

\(9-32\) n Determine whether the sequence converges or diverges. If it converges, find the limit. $$a_{n}=\cos (2 / n)$$

Step-by-Step Solution

Answer

The sequence converges to 1.

Step 1: Understand the Sequence

The sequence given is \( a_n = \cos(2/n) \). This sequence involves splitting the number 2 by \( n \) and finding the cosine of the resulting value.

Step 2: Analyze the Sequence as n Approaches Infinity

As \( n \) approaches infinity, the term \( 2/n \) approaches zero since any non-zero constant divided by an infinitely large number becomes zero. We need to consider the behavior of the cosine function as its argument approaches zero.

Step 3: Determine the Limit of the Function

The cosine function, \( \cos(x) \), approaches 1 as \( x \to 0 \). Thus, as \( n \to \infty \), \( \cos(2/n) \to \cos(0) = 1 \).

Step 4: Conclude about Convergence or Divergence

Since the limit \( \lim_{{n \to \infty}} a_n = 1 \), the sequence converges to 1.

Key Concepts

Limit of a Sequence

When we talk about the 'limit of a sequence,' we're essentially trying to discover what value the terms of a sequence get closer to as the number of terms increases. Imagine a sequence like stepping stones leading to a particular point. As you step further along, you get closer and closer to that point. In this particular exercise with the sequence \( a_n = \cos(2/n) \), we're interested in what happens to this sequence as \( n \), the number of terms, becomes very large.To find the limit, we observe what happens to the expression inside the cosine function. The term \( 2/n \) behaves such that as \( n \) becomes large (approaching infinity), \( 2/n \) shrinks towards zero. The cosine function, which takes this ever-smaller number, will thus approach the value of \( \cos(0) \). Therefore, since \( \cos(0) \) equals 1, the limit of the sequence \( a_n \) is 1. This means that no matter how large \( n \) grows, the terms of the sequence will get closer and closer to 1.

Behavior as n Approaches Infinity

Understanding what happens as \( n \) approaches infinity is crucial in calculus and understanding sequences. In this context, \( n \) represents the term number of the sequence. When we say \( n \rightarrow \infty \), we're investigating the long-term behavior of the sequence. In simple terms, when \( n \) is very large, the fraction \( 2/n \) becomes very small. It's like dividing a pie (number 2) into many, many slices (denoted by \( n \)); each slice becomes tiny as \( n \) increases. When this tiny slice is plugged into the cosine function, \( \cos(2/n) \), it mimics the behavior of \( \cos(0) \) since \( 2/n \) is nearly zero.Thus, while each term in the sequence adjusts based on the size of \( n \), its pathway stabilizes towards a steady outcome — the value of 1. It's like a river flowing towards a single ocean point no matter the twists and turns along its way.

Cosine Function

The cosine function, often denoted as \( \cos(x) \), is a fundamental concept in trigonometry. It describes a wave-like pattern and is crucial for understanding periodic behaviors. The function starts at a maximum value of 1 when \( x = 0 \), which means that if the input is zero, the output of the cosine function is 1.A unique property of the cosine function is how it behaves when its input is very small or very large:

• Cosine of zero: \( \cos(0) = 1 \)
• For small numbers: As \( x \to 0 \), \( \cos(x) \) approaches 1.
• For large numbers: The function repeats every \( 2\pi \).

This is why, in the given sequence \( a_n = \cos(2/n) \), as \( n \) rises, and \( 2/n \) closes in on zero, the sequence results gravitate around 1. Thus, the cosine function plays a critical role in evaluating limits of sequences that involve angles or cyclical behavior, such as the one we find in trigonometric series. The cosine function helps us comprehend how sequences stabilize towards their limits with growing term numbers.