Problem 51 - Calculus: Early Transcendentals

Question

Determine whether the sequence converges or diverges. If it converges, find the limit. \( a_n = \arctan (\ln n) \)

Step-by-Step Solution

Answer

The sequence converges to \( \frac{\pi}{2} \).

Step 1: Understanding the Sequence

We are given a sequence \( a_n = \arctan (\ln n) \). The task is to determine whether this sequence converges or diverges and to find the limit if it converges.

Step 2: Look at the Behavior of \( \ln n \)

Consider the function \( \ln n \) as \( n \to \infty \). The natural logarithm \( \ln n \) increases without bound, meaning \( \ln n \to \infty \) as \( n \to \infty \).

Step 3: Behavior of \( \arctan(x) \) as \( x \to \infty \)

Now consider \( \arctan(x) \). As \( x \to \infty \), \( \arctan(x) \to \frac{\pi}{2} \). This is because the arctangent function approaches \( \frac{\pi}{2} \) asymptotically when its input goes to infinity.

Step 4: Determine Limit of \( a_n \)

Since \( \ln n \to \infty \) implies that \( a_n = \arctan(\ln n) \to \arctan(\infty) = \frac{\pi}{2} \), we determine that the sequence \( a_n \) converges to \( \frac{\pi}{2} \).

Key Concepts

Understanding the Arctan Function

The arctan function, short for the arctangent function, is the inverse of the tangent function. This means that for any angle \( \theta \), \( \arctan(\tan(\theta)) = \theta \). While the tangent function can extend infinitely in both directions, the output of the arctan function is restricted. It limits its values to a range from \(-\frac{\pi}{2} \) to \( \frac{\pi}{2} \), making it quite unique.

• As the input of arctan, which we often call \( x \), grows larger and larger (heading towards infinity), the output \( \arctan(x) \) approaches \( \frac{\pi}{2} \).
• This behavior is described as asymptotic because arctan gets closer and closer to \( \frac{\pi}{2} \) but never actually reaches it.

The arctan function gradually increases with larger positive values and decreases with larger negative values.Understanding these properties is crucial when considering sequences involving the arctan function, as it helps in establishing how such sequences behave at infinity.

The Concept of Limits

In calculus, limits are foundational in understanding how functions behave as their inputs approach a certain value. In this specific example, we're determining the behavior of the sequence \( a_n = \arctan(\ln n) \) as \( n \to \infty \).When we say the sequence converges, we mean that as \( n \) gets larger and larger, the value of \( a_n \) approaches a particular number.

• This number is known as the limit of the sequence.
• If we say it diverges, it means no such single number exists, to which the sequence values approach.

Limits are crucial because they allow us to predict long-term behavior of sequences and functions, even if the direct computation of these values seems complex or daunting. Consider limits as a way to "zoom out" and see the "big picture" of how a function or sequence behaves over a large scale.

Exploring the Natural Logarithm

The natural logarithm function, denoted as \( \ln(n) \), provides insights on how quickly values grow or shrink. Unlike other logarithms, the natural logarithm is based on the mathematical constant \( e \), approximately equal to 2.718.

• One of the key properties of \( \ln(n) \) is that as \( n \to \infty \), \( \ln(n) \) also increases towards infinity — albeit more slowly than linear functions.
• This implies that \( \ln(n) \) will never stop growing and does not approach any finite number.

When analyzing mathematical problems and sequences, distinguishing the growth rates of functions like \( \ln(n) \) helps clarify how other functions or sequences involving the logarithm will behave. Knowing that \( \ln(n) \to \infty \) aids us in concluding that the input to the arctan in our sequence will also head towards infinity, allowing the sequence \( a_n = \arctan(\ln n) \), consequently, to approach \( \frac{\pi}{2} \).