Question

Determine whether the sequence with th given \(n\) th term is monotonic. Discuss the boundedness of th sequence. Use a graphing utility to confirm your results. \(a_{n}=\sin \frac{n \pi}{6}\)

Short answer

The given sequence is non-monotonic as it does not either solely increase or decrease and the sequence is bounded as all terms lie between -1 and 1.

Step-by-step solution

Identify the sequence

The given sequence is \(a_{n}=\sin{\frac{n\pi}{6}}\). We can evaluate first few terms of the sequence to observe the pattern: \(a_{1}\) = \(\sin{\frac{\pi}{6}} = 0.5, a_{2} = \sin{\frac{\pi}{3}} = √3/2, a_{3} =\sin0.5 = 0.5, a_{4} =\sin{\frac{2\pi}{3}} = √3/2\) and so on.

Determine the monotonicity

As evaluated in step 1, the first few terms of the sequence are neither solely increasing nor decreasing. It oscillates periodically between 0.5 and √3/2. Therefore, the sequence is not monotonic.

Discuss the boundedness

The values of a sequence generated by a sine function always lie between -1 and 1. Thus, this sequence is bounded above by 1 and below by -1.

Confirm results using a graphing utility

You can confirm the findings by plotting the terms of the sequence using a graphing utility where the y-coordinate is \(a_{n}\) and the x-coordinate is \(n\). The graph will show a periodically oscillating pattern confirming the sequence is not monotonic. The peaks and valleys of the graph will confirm the sequence is bounded between -1 and 1.

Key concepts

Bounded Sequence

A sequence is considered to be bounded if its terms are confined within a fixed set of limits. In more formal terms, a sequence \( \{a_n\} \) is said to be bounded above if there exists a number M such that for all n, \( a_n \leq M \). Similarly, a sequence is bounded below if there exists a number m so that for all n, \( a_n \geq m \). If a sequence is both bounded above and below, it is simply said to be bounded.

In the given exercise, the sequence \( a_n = \sin\left(\frac{n\pi}{6}\right) \) is bounded because the sine function only outputs values between -1 and 1. This inherent characteristic of the sine function guarantees that no matter the value of n, the sequence's terms will not exceed these bounds, confirming the boundedness of the sequence. Utilizing a graphing utility can help visually confirm this property, as the output values plotted would all be within the [-1,1] interval.

Sequence Oscillation

The concept of oscillation in sequences occurs when the terms of the sequence do not settle into a pattern that is either strictly increasing or decreasing, but instead vary up and down over a range of values. An oscillating sequence, such as \( a_n = \sin\left(\frac{n\pi}{6}\right) \) from the exercise, does not adhere to typical monotonic behavior.

Sequences derived from trigonometric functions like sine and cosine are classic examples that demonstrate oscillatory behavior. The values of \( a_n \) oscillate between 0.5 and \( \sqrt{3}/2 \) in a periodic fashion. This implies that the sequence lacks a consistent direction of change and instead repeats its values in a cycle. It's important to note that the amplitude of these oscillations is contained within the bounded range of the function, which in the case of the sine function is [-1,1].

Graphing Utility Mathematics

Graphing utilities are incredibly valuable tools in mathematics, especially for visualizing sequences and their characteristics. These utilities can plot the terms of a sequence against their indices, revealing patterns or properties that might be less apparent from the sequence's formula alone.

In our exercise, using a graphing utility to plot \( a_n \) against n would show a wave-like pattern, mapping out the oscillating nature of the sequence. The graph would demonstrate how the sequence values peak at 1 and trough at -1, confirming the sequence is bounded. Furthermore, the non-monotonic nature of \( a_n \) becomes clear, as the terms do not progress in a single direction but rise and fall periodically. This visual aid can substantiate a student's understanding of sequences in a powerful way and reinforce the insights obtained through analytical means.