Textbook Question
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{2ⁿ⁺¹3⁻ⁿ}
63
views
14. Sequences & Series
Sequences
1:11 minutesProblem 10.2.35
Textbook Question
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{tan⁻¹(n)}
Verified step by step guidance1
Recognize that the sequence is given by \( a_n = \tan^{-1}(n) \), where \( \tan^{-1} \) is the inverse tangent function, also known as arctangent.
Recall the behavior of the arctangent function: as its input \( x \) approaches infinity, \( \tan^{-1}(x) \) approaches a horizontal asymptote at \( \frac{\pi}{2} \).
Since \( n \) is a sequence that increases without bound (\( n \to \infty \)), consider the limit \( \lim_{n \to \infty} \tan^{-1}(n) \).
Use the known limit property: \( \lim_{x \to \infty} \tan^{-1}(x) = \frac{\pi}{2} \). This implies the sequence \( a_n = \tan^{-1}(n) \) converges to \( \frac{\pi}{2} \).
Conclude that the limit of the sequence \( \{ \tan^{-1}(n) \} \) as \( n \to \infty \) is \( \frac{\pi}{2} \).
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limits of Sequences
The limit of a sequence is the value that the terms of the sequence approach as the index goes to infinity. If the terms get arbitrarily close to a specific number, the sequence converges to that limit; otherwise, it diverges.
Recommended video:
Guided course
8:22
Introduction to Sequences
Inverse Tangent Function (arctan)
The inverse tangent function, arctan(x), returns the angle whose tangent is x. It is defined for all real numbers and has horizontal asymptotes at ±π/2, meaning as x approaches ±∞, arctan(x) approaches ±π/2.
Recommended video:
3:17Inverse Tangent
Behavior of arctan(n) as n → ∞
As the input n grows without bound, arctan(n) approaches its horizontal asymptote π/2. This means the sequence {arctan(n)} converges to π/2, since the values get closer and closer to this limit for large n.
Recommended video:
Guided course
5:52Writing a General Formula Example 1
Related Videos
Related Practice