Problem 81 - 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. $$ a_{n}=\tan ^{-1} n $$

Step-by-Step Solution

Answer

The sequence converges with a limit of \(\pi/2\).

Step 1: Understand the Function

The function we are working with is the inverse tangent, or arctangent, function, represented by \( \tan^{-1}(x) \). This function outputs an angle whose tangent is the input \( x \). The range of \( \tan^{-1}(x) \) is \((-\pi/2, \pi/2)\).

Step 2: Analyze the Sequence Behavior

Consider the sequence \( a_n = \tan^{-1}(n) \). As \( n \) increases, \( n \to \infty \). We need to analyze the behavior of \( \tan^{-1}(n) \) as \( n \) grows larger.

Step 3: Determine the Limit

As \( n \to \infty \), \( \tan^{-1}(n) \to \pi/2 \), because the arctangent function approaches \( \pi/2 \) when the input becomes very large. Therefore, the limit of the sequence \( a_n = \tan^{-1}(n) \) is \( \pi/2 \).

Step 4: Convergence Conclusion

Since the sequence \( a_n = \tan^{-1}(n) \) approaches the finite value \( \pi/2 \), the sequence converges.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are essential tools in mathematics. Specifically, the function \( \tan^{-1}(x) \) is known as arctangent. It helps us find the angle whose tangent equals \( x \). Think of it as reversing the tangent function. While the tangent function has values that range across all real numbers, the arctangent works differently.

Its range is constrained to \((-\pi/2, \pi/2)\). This means the output of the \( \tan^{-1}(x) \) function always lies within these boundaries. This property makes it unique among other inverse trigonometric functions. Understanding these functions is crucial, as they allow for conversion between multiple forms of angle measurement.

• Specifically, \( \tan^{-1}(x) \) stabilizes around \( \pi/2 \) as \( x \) grows large, and \(-\pi/2 \) as \( x \) decreases significantly.
• It's essential to realize that as \( n \) approaches infinity, \( \tan^{-1}(n) \) will steadily climb towards \( \pi/2 \).

Limits of Sequences

Limits are at the heart of understanding sequences and their long-term behavior. When we evaluate a sequence like \( a_n = \tan^{-1}(n) \), we look at its limit as \( n \) approaches infinity. A limit tells us where the terms of a sequence are heading, even if they never quite "arrive" there.

For \( a_n = \tan^{-1}(n) \), as \( n \to \infty \), this sequence trends towards a specific limit. This limit is \( \frac{\pi}{2} \). Why \( \frac{\pi}{2} \)? Because, mathematically, the function \( \tan^{-1}(x) \) moves closer and closer to \( \pi/2 \) as its input \( x \) becomes very large.

• The concept of limits helps form the foundation of calculus, offering the ability to predict a sequence's behavior at infinity.
• Determining limits is often the step needed to establish whether sequences converge or diverge.

Sequence Behavior

The behavior of a sequence gives insights into how its terms behave as \( n \) grows. Consider the sequence \( a_n = \tan^{-1}(n) \). Here, understanding the behavior involves watching how terms of the sequence change as \( n \to \infty \).

In simple terms, the main feature of a sequence is whether it tends towards a limit or not.

• A sequence is said to **converge** if it approaches a finite limit, as \( a_n \) does towards \( \pi/2 \).
• If no such limit exists, it **diverges**.
• In our case, \( \tan^{-1}(n) \) provides a clear example of convergence, slowly moving towards the fixed point of \( \pi/2 \).

Observing a sequence's behavior helps us evaluate its characteristics and predict its long-term trends.