Question

Determine whether the sequence is bounded, bounded above, bounded below, or none of the above. $$\left\\{a_{n}\right\\}=\\{\tan n\\}$$

Short answer

The sequence \( \{a_n\} = \{\tan n\} \) is neither bounded above nor below.

Step-by-step solution

Understanding the Sequence

The given sequence is defined by \(a_n = \tan n\), where \(n\) is a natural number (1, 2, 3,...). The tangent function, \(\tan x\), has a periodic behavior with periodic intervals of \(\pi\).

Analyzing the Tangent Function

The function \(\tan x\) is periodic with a period of \(\pi\) and has vertical asymptotes at \(x = \frac{\pi}{2} + k\pi\), where \(k\) is an integer. This means \(\tan x\) will approach infinity as \(x\) approaches these points.

Considering Boundedness Above and Below

For \(a_n = \tan n\) to be bounded, there must be real numbers \(M\) (upper bound) and \(m\) (lower bound) such that \(m \leq \tan n \leq M\) for all natural numbers \(n\).

Observing the Sequence Behavior

Due to the periodic nature of the tangent function and the vertical asymptotes at regular intervals, \(\tan n\) will take on arbitrarily large positive and negative values as \(n\) approaches values near \(\frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2},\) etc.

Conclusion on Boundedness

Since \(\tan n\) can take on arbitrarily large positive and negative values, the sequence \(\{a_n\} = \{\tan n\}\) is neither bounded above nor bounded below. Thus, the sequence is unbounded.

Key concepts

Bounded Sequences

In mathematics, a sequence is said to be bounded if there exists a real number that serves as a limit that the sequence's terms do not exceed. There are two types of boundedness to consider: bounded above and bounded below.

• A sequence is bounded above if there is a maximum value, or upper limit, that no term in the sequence surpasses.
• Similarly, a sequence is bounded below if there is a minimum value, or lower limit, that no term drops below.

In terms of the sequence given in the problem, \(a_{n} = an n\), the task is to determine these characteristics. It must be shown whether or not there exist ultimate limits on both the positive and negative ends of this sequence. Understanding boundedness is crucial in comprehending the behavior and stability of sequences.

Tangent Function

The tangent function, denoted as \(\tan x\), is a fundamental trigonometric function that arises frequently in mathematical equations and real-world phenomena. Unlike the sine and cosine functions, the tangent function has a unique property—it is undefined at specific points and takes on infinite values at others.

• The function has vertical asymptotes at \(x = \frac{\pi}{2} + k\pi\), where \(k\) represents any integer.
• At these asymptotes, the function's value approaches infinity, making it undefined at those specific points.

Additionally, because \(\tan x\) is derived from the sine and cosine functions (being equal to \(\frac{\sin x}{\cos x}\)), its value changes dramatically as \(x\) reaches the points where \(\cos x = 0\). Therefore, understanding the properties of the tangent function is essential for analyzing any sequence defined using it, such as \(a_{n} = \tan n\).

Periodic Behavior

Periodic behavior refers to the repeating patterns seen in functions or sequences across specific intervals. The tangent function showcases this characteristic with its indefinite repetition over time, known as a period.

• The period of the tangent function is \(\pi\), meaning that it repeats every \(\pi\) units along the x-axis.
• This behavior results in a predictable, cyclical pattern in the values of \(\tan x\).

However, despite its periodicity, \(\tan x\) also contains points where its value is not constrained—it grows unbounded. As a result, when investigating sequences like \(a_{n} = \tan n\), we observe that even though there is a repeating nature, the amplitude and direction of the values can vary dramatically near its asymptotes, leading to unbounded behavior. Understanding periodic behavior and its implications helps in analyzing the vibrancy and stability of functions and sequences.