Problem 63 - Essential Calculus

Question

In Exercises \(63-72,\) determine whether the sequence with the given \(n\) th term is monotonic. Discuss the boundedness of the sequence. Use a graphing utility to confirm your results. \(a_{n}=\frac{n}{2^{n+2}}\)

Step-by-Step Solution

Answer

The sequence \(a_{n}=\frac{n}{2^{n+2}}\) is monotonic and bounded. It is non-increasing or monotonic since the difference between any two consecutive terms consistently decreases. It is bounded by 0 and \(\frac{1}{8}\) as every term of the sequence is less than or equal to \(\frac{1}{8}\).

Step 1: Check if the Sequence is Monotonic

A sequence is monotonic if it's either entirely non-increasing or non-decreasing. To check this we'll examine the difference between a term at position (n+1) and a term at position n. If the value is consistently decreasing, then the sequence is non-increasing, if it's consistently increasing then the sequence is non-decreasing. So we need to calculate \( \frac{n+1}{2^{n+3}} - \frac{n}{2^{n+2}} = \frac{2n + 2 - 2n}{2^{n+3}} = \frac{2}{2^{n+3}}\). Since n can be any positive number, the value will consistently decrease, showing the sequence is non-increasing or monotonic.

Step 2: Check if the Sequence is Bounded

A sequence is bounded if there is a real number M such that every term of the sequence is less than or equal to M. Since the numerator of the sequence grows linearly and the denominator grows exponentially, the sequence values will get smaller each step. Therefore, the sequence is bounded by 0 on the left (smaller side), and the starting term \(\frac{1}{8}\) on the right (larger side). So we can certainly say that every term of the sequence will be less than or equal to \(\frac{1}{8}\), thus it is bounded.

Step 3: Use a Graphing Utility for Visual Confirmation

Plotting the sequence using a graphing utility should show that the sequence is decreasing (from examination of the graph slope) and gets closer to zero with increasing n (indicating that the sequence is bounded). Important to note that physical plotting isn't necessary for solving this problem but a useful skill to interpret and validate the results.

Key Concepts

Bounded Sequence

Understanding what makes a sequence bounded can be the key to solving many mathematical challenges. A sequence is defined as bounded when its terms do not exceed certain limits, both upper and lower. This implies there exists a real number, say \(M\), such that every term of the sequence is less than or equal to \(M\). Similarly, there should also be a lower bound \(m\) such that every term is greater than or equal to \(m\). For example, consider our sequence with the general term defined by \(a_{n} = \frac{n}{2^{n+2}}\).

• The numerator \(n\) increases linearly.
• The denominator \(2^{n+2}\) grows exponentially.

Due to the faster growth of the denominator, the value of the sequence's terms decreases as \(n\) increases. This indicates the sequence has upper and lower bounds, 0 and \(\frac{1}{8}\), respectively. Thus, the sequence is bounded.

Graphing Utility

Using a graphing utility is a powerful method to visually confirm the behavior or nature of a mathematical sequence or function. It allows you to plot terms and analyze trends like monotonicity or boundedness. Here’s how you can utilize a graphing utility effectively for our sequence \(a_{n} = \frac{n}{2^{n+2}}\):

• Plot the sequence terms over a suitable range for \(n\).
• Observe the curve's slope to verify monotonicity – whether it's consistently increasing or decreasing.
• Examine how close the sequence terms approach a value, such as zero, to check boundedness.

While graphing may not always be necessary for basic homework solutions, it's invaluable for exploring deeper insights. Visual patterns can aid in understanding theoretical findings, confirming calculations, and identifying trends that may be hard to see in equations alone.

Exponential Growth

Exponential growth is a common phenomenon in mathematics, science, and nature. When a quantity increases quickly with its value at each step being a constant multiple of the previous one, it is said to grow exponentially. In our sequence \(a_{n} = \frac{n}{2^{n+2}}\), the denominator \(2^{n+2}\) illustrates this concept.

• As \(n\) increases, the denominator grows by a factor of 2 for each increment in \(n\).
• This rapid increase in the denominator causes the overall value of \(a_n\) to decrease, despite the linear increase in the numerator \(n\).

Understanding exponential growth helps in analyzing sequences since it greatly influences how terms behave and establish whether the sequence is converging or diverging. The balance between linear and exponential terms will often dictate the sequence's boundedness and monotonicity.