Problem 40 - Calculus: Early Transcendentals

Question

Determine whether the sequence converges or diverges. If it converges, find the limit. \( a_n = \frac {\tan^{-1}n}{n} \)

Step-by-Step Solution

Answer

The sequence converges, and its limit is 0.

Step 1: Understand the Sequence

The sequence given is \( a_n = \frac{\tan^{-1}(n)}{n} \). This is an expression dependent on \( n \), where the numerator involves the inverse tangent function and the denominator is \( n \).

Step 2: Analyze the Behavior of \( \tan^{-1}(n) \)

As \( n \to \infty \), the function \( \tan^{-1}(n) \) approaches \( \frac{\pi}{2} \). This is because the tangent inverse of very large numbers is approximately \( \frac{\pi}{2} \).

Step 3: Determine the Limit of the Sequence

Consider the sequence limit: \[ \lim_{{n \to \infty}} a_n = \lim_{{n \to \infty}} \frac{\tan^{-1}(n)}{n} \]We know that \( \tan^{-1}(n) \to \frac{\pi}{2} \) but since the denominator \( n \to \infty \), the overall fraction approaches 0.Thus,\[ \lim_{{n \to \infty}} \frac{\tan^{-1}(n)}{n} = 0 \]

Step 4: Conclusion on Convergence

Since the limit of the sequence \( \lim_{{n \to \infty}} a_n = 0 \), the sequence converges to 0.

Key Concepts

Limit of a Sequence

In mathematics, the limit of a sequence is the value that the terms of a sequence "approach" as the index (often denoted by \( n \)) goes to infinity. It's like watching an object get very close to a point but never quite touching it. Understanding this concept is crucial because it tells us about the behavior of sequences over long periods of progression. For any given sequence \( a_n \), we say that \( a_n \) converges if it approaches a specific number \( L \) when \( n \rightarrow \infty \). If no such \( L \) exists, then the sequence diverges. One common technique to determine convergence is to directly compute the limit:

• If \( \lim_{{n \to \infty}} a_n = L \), the sequence converges to \( L \).
• If \( \lim_{{n \to \infty}} a_n \) does not equal a specific value, the sequence diverges.

Inverse Trigonometric Functions

Inverse trigonometric functions are the inverse functions of the trigonometric functions, used to obtain an angle from the ratios of sides of a right-angled triangle. For example, \( \tan^{-1}(x) \) gives you the angle whose tangent is \( x \). For large values of \( x \), \( \tan^{-1}(x) \) behaves predictably:

• It approaches \( \frac{\pi}{2} \) as \( x \to \infty \). This means that as the value of \( x \) increases, the angle approaches 90 degrees.
• Conversely, \( \tan^{-1}(x) \) approaches \( -\frac{\pi}{2} \) as \( x \to -\infty \).

These behaviors help in understanding how inverse trigonometric functions contribute to the limit of sequences that involve them. Their limiting behavior, as shown in sequences like \( a_n = \frac{\tan^{-1}(n)}{n} \), can be crucial to determining the sequence's convergence.

Infinite Limit Analysis

Infinite limit analysis involves evaluating what happens to a function or sequence as its input or index goes to infinity. This type of analysis is key for understanding sequences and series in calculus. By finding the limit at infinity, you can determine whether a sequence converges.When dealing with sequences like \( a_n = \frac{\tan^{-1}(n)}{n} \), we analyze how both the numerator and the denominator behave separately as \( n \to \infty \).

• **Numerator Analysis:** The term \( \tan^{-1}(n) \) approaches \( \frac{\pi}{2} \).
• **Denominator Analysis:** The term \( n \) simply grows without bound.

By comparing these, since the numerator becomes almost a constant and the denominator grows infinitely, the whole fraction \( \frac{\pi/2}{n} \) shrinks towards zero. These independent behaviors, when examined closely, help conclude that some sequences converge, as they do in the provided example.