Problem 47 - Thomas Calculus

Question

Which of the sequences \(\left\\{a_{n}\right\\}\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\sin \left(\frac{\pi}{2}+\frac{1}{n}\right) $$

Step-by-Step Solution

Answer

The sequence converges to 1.

Step 1: Analyze the Sequence Behavior at Limit

Consider the sequence \(a_n = \sin \left(\frac{\pi}{2} + \frac{1}{n}\right)\). As \(n\) approaches infinity, \(\frac{1}{n}\) approaches 0. This means the expression inside the sine function approaches \(\frac{\pi}{2}\).

Step 2: Determine the Limit of the Sine Function

Since the function inside the sine approaches \(\frac{\pi}{2}\), and \(\sin(\frac{\pi}{2}) = 1\), the sequence \(a_n\) should also approach 1 as \(n\) approaches infinity.

Step 3: Convergence Conclusion

As the sequence approaches a fixed number \(1\), \(a_n = \sin \left(\frac{\pi}{2} + \frac{1}{n}\right)\) converges to \(1\).

Key Concepts

Understanding the Limit of a Sequence

The concept of a limit of a sequence is an important tool in calculus and real analysis. It helps us determine what value, if any, the terms of a sequence approach as the sequence progresses indefinitely. When we talk about the limit of a sequence, we're asking what number the terms of the sequence are getting closer to as the sequence continues.The notation \(\lim_{{n \to \infty}} a_n = L\) means that as \(n\) becomes very large, the terms \(a_n\) of the sequence get closer and closer to the number \(L\). In simpler terms, the sequence "settles down" to the number \(L\). For the sequence given in the exercise, the limit is determined by analyzing the behavior of \(\sin(\pi/2 + 1/n)\) as \(n\) approaches infinity.

The Role of the Sine Function

The sine function is a periodic trigonometric function that oscillates between -1 and 1. Within this range, different inputs to the sine function produce different outputs between these bounds. The angle measurement we input, whether in degrees or radians, determines where we are on the wave that the sine function creates.In the context of our sequence \(a_n = \sin \left(\frac{\pi}{2} + \frac{1}{n}\right)\), as \(n\) increases, \(1/n\) gets smaller, pushing the argument of the sine function closer to \(\pi/2\). Understanding how the sine function behaves around this value is crucial, because \(\sin(\pi/2) = 1\). Thus, as the input \(\frac{\pi}{2} + \frac{1}{n}\) inches closer to \(\pi/2\), the sine function produces values closer to 1.

Infinity Behavior and its Implications

Infinity in mathematics isn't a number, but a concept that describes something without bound. It's something that keeps going without ever stopping. When we talk about \(n\) approaching infinity in the context of sequences, we're considering what happens to the sequence's behavior when \(n\) increases without limit.In our particular sequence \(a_n = \sin \left(\frac{\pi}{2} + \frac{1}{n}\right)\), as \(n\) heads towards infinity, the term \(1/n\) approaches 0. This suggests that \(\frac{\pi}{2} + \frac{1}{n}\) gets ever closer to \(\pi/2\). Therefore, knowledge of infinity helps us understand that the behavior of sequences can be predictable even if their terms are initially varied.

Identifying a Convergent Sequence

A convergent sequence is one where the terms of the sequence approach a single fixed value as the sequence progresses towards infinity. Determining convergence involves checking the existence of a limit.In our example, the sequence \(a_n = \sin \left(\frac{\pi}{2} + \frac{1}{n}\right)\) converges because its terms get closer and closer to 1. As we found by examining the limit and using the properties of trigonometric functions, the terms approach the constant value \(\sin(\pi/2)\) as \(\frac{1}{n}\) shrinks to 0. This convergence means that no matter how far out you go in the sequence, the terms will be extremely close to 1, illustrating a settled behavior. Thus, this sequence is a perfect example of convergence in action.